denoted by p(event) this method for calculating probabilities is only appropriate when the outcomes...
TRANSCRIPT
Denoted by P(Event)
outcomes total
outcomes favorable)( EP
This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely
The relative frequency at which a chance experiment occursFlip a fair coin 30 times amp get 17 heads
30
17
As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Rule 1 Legitimate ValuesFor any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs A randomly selected student is female - What
is the probability she plays soccer for FHS
A randomly selected student is female - What is the probability she plays football for FHS
Independent
Not independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
)( BAP Independent
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
)( BAP Independent
Yes No
)()( BPAP )()|( APABP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
The relative frequency at which a chance experiment occursFlip a fair coin 30 times amp get 17 heads
30
17
As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Rule 1 Legitimate ValuesFor any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs A randomly selected student is female - What
is the probability she plays soccer for FHS
A randomly selected student is female - What is the probability she plays football for FHS
Independent
Not independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
)( BAP Independent
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
)( BAP Independent
Yes No
)()( BPAP )()|( APABP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
As the number of repetitions of a chance experiment increase the difference between the relative frequency of occurrence for an event and the true probability approaches zero
Rule 1 Legitimate ValuesFor any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs A randomly selected student is female - What
is the probability she plays soccer for FHS
A randomly selected student is female - What is the probability she plays football for FHS
Independent
Not independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
)( BAP Independent
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
)( BAP Independent
Yes No
)()( BPAP )()|( APABP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Rule 1 Legitimate ValuesFor any event E 0 lt P(E) lt 1
Rule 2 Sample spaceIf S is the sample space P(S) = 1
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs A randomly selected student is female - What
is the probability she plays soccer for FHS
A randomly selected student is female - What is the probability she plays football for FHS
Independent
Not independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
)( BAP Independent
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
)( BAP Independent
Yes No
)()( BPAP )()|( APABP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Rule 3 Complement
For any event E P(E) + P(not E) = 1
Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs A randomly selected student is female - What
is the probability she plays soccer for FHS
A randomly selected student is female - What is the probability she plays football for FHS
Independent
Not independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
)( BAP Independent
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
)( BAP Independent
Yes No
)()( BPAP )()|( APABP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs A randomly selected student is female - What
is the probability she plays soccer for FHS
A randomly selected student is female - What is the probability she plays football for FHS
Independent
Not independent
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
)( BAP Independent
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
)( BAP Independent
Yes No
)()( BPAP )()|( APABP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Rule 4 Multiplication
If two events A amp B are independent
General rule
P(B) P(A) B) ampP(A
A)|P(B P(A) B) ampP(A
)( BAP
)( BAP Independent
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
)( BAP Independent
Yes No
)()( BPAP )()|( APABP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
)( BAP
)( BAP Independent
Yes
)()( BPAP
What does this mean
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
)( BAP Independent
Yes No
)()( BPAP )()|( APABP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Ex 1) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that both bulbs are defective
Can you assume they are independent
00250505D) amp P(D
)( BAP
)( BAP Independent
Yes No
)()( BPAP )()|( APABP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
)( BAP
)( BAP Independent
Yes No
)()( BPAP )()|( APABP
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Ex 2) There are seven girls and eight boys in a math class The teacher selects two students at random to answer questions on the board What is the probability that both students are girlsAre these events independent
2146
157
G) amp P(G
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Ex 6) Suppose I will pick two cards from a standard deck without replacement What is the probability that I select two spades
Are the cards independent NO
P(A amp B) = P(A) P(B|A)
Read ldquoprobability of B given that A occursrdquo
P(Spade amp Spade) = 14 1251 = 117
The probability of getting a spade given that a spade has already been drawn
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Rule 5 AdditionIf two events E amp F are disjoint
P(E or F) = P(E) + P(F)
(General) If two events E amp F are not disjoint
P(E or F) = P(E) + P(F) ndash P(E amp F)
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
)( FEP What does this mean
)( FEP Mutually exclusive
)()( FPEP
Yes
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Ex 3) A large auto center sells cars made by many different manufacturers Three of these are Honda Nissan and Toyota Suppose that P(H) = 25 P(N) = 18 P(T) = 14 What is the probability that the next car sold is a Honda Nissan or Toyota
Are these disjoint events
P(H or N or T) =
P(not (H or N or T) =
yes
25 + 18+ 14 = 57
1 - 57 = 43
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Ex 4) Musical styles other than rock and pop are becoming more popular A survey of college students finds that the probability they like country music is 40 The probability that they liked jazz is 30 and that they liked both is 10 What is the probability that they like country or jazz
P(C or J) = 4 + 3 -1 = 6
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
)( FEP
)( FEP Mutually exclusive
)()( FPEP
Yes No
)()()( FEPFPEP
Independent Yes )()( FPEP
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Ex 5)
If P(A) = 045 P(B) = 035 and A amp B are independent find P(A or B)Is A amp B disjoint
If A amp B are disjoint are they independent
Disjoint events do not happen at the same time
So if A occurs can B occur
Disjoint events are dependent
NO independent events cannot be disjoint
P(A or B) = P(A) + P(B) ndash P(A amp B)
How can you find the
probability of A amp B
P(A or B) = 45 + 35 - 45(35) = 06425
If independent
multiply
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6
If a coin is flipped amp a die rolled at the same time what is the probability that you will get a tail or a number less than 3
T1
T2
T3
T4
T5
T6 H1
H3
H2
H5
H4
H6
Coin is TAIL Roll is LESS THAN 3
P (heads or even) = P(tails) + P(less than 3) ndash P(tails amp less than 3)
= 12 + 13 ndash 16= 23
Flipping a coin and rolling a die are independent events
Independence also implies the events are NOT disjoint (hence the overlap)
Count T1 and T2 only once
Sample Space
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Ex 7) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that exactly one bulb is defective
P(exactly one) = P(D amp DC) or P(DC amp D)
= (05)(95) + (95)(05)
= 095
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Ex 8) A certain brand of light bulbs are defective five percent of the time You randomly pick a package of two such bulbs off the shelf of a store What is the probability that at least one bulb is defective
P(at least one) = P(D amp DC) or P(DC amp D) or (D amp D)
= (05)(95) + (95)(05) + (05)(05)
= 0975
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
A probability that takes into account a given condition
P(A)
B)P(AA)|P(B
P(given)
P(and)A)|P(B
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Ex 10) In a recent study it was found that the probability that a randomly selected student is a girl is 51 and is a girl and plays sports is 10 If the student is female what is the probability that she plays sports
196151
1
P(F)
F)P(S F)|P(S
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Ex 11) The probability that a randomly selected student plays sports if they are male is 31 What is the probability that the student is male and plays sports if the probability that they are male is 49
1519
4931
P(M)
M)P(SM)|P(S
x
x
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
359195
)( StudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
12) What is the probability that the driver is a student
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
35945
)( EuropeanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
13) What is the probability that the driver drives a European car
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
359102212
)(
AsianorAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
14) What is the probability that the driver drives an American or Asian car
Disjoint
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
35947102164
)(
AsianorStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
15) What is the probability that the driver is staff or drives an Asian car
Disjoint
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
35947
)( AsianandStaffP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
16) What is the probability that the driver is staff and drives an Asian car
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
195107
)|( StudentAmericanP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
17) If the driver is a student what is the probability that they drive an American car
Condition
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
4533
)|( EuropeanStudentP
Stu Staff TotalAmerican 107 105 212European 33 12 45Asian 55 47 102Total 195 164 359
18) What is the probability that the driver is a student if the driver drives a European car
Condition
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Example 19Management has determined that customers return 12 of the items assembled by inexperienced employees whereas only 3 of the items assembled by experienced employees are returned Due to turnover and absenteeism at an assembly plant inexperienced employees assemble 20 of the items Construct a tree diagram or a chart for this data
What is the probability that an item is returned
If an item is returned what is the probability that an inexperienced employee assembled it
P(returned) = 48100 = 0048
P(inexperienced|returned) = 2448 = 05
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100
Returned Not returned
Total
Experienced 24 776 80
Inexperienced 24 176 20
Total 48 952 100