session 1 2014 version
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Introduction to Probability
RemalynQ. Casem
email: [email protected]
2014 1-1Probability Theory
2014 Probability Theory 1-2
Sample space and events
TOPICS
Axioms of probability
Finite probability spaces
Finite equiprobable spaces
Set and event operations
Properties of probability
Experiment
2014 1-3
Probability Theory
Sample Space and Events
Sample space
Outcome
a procedure that generatesobservable outcomes
any possible observation of anexperiment.
mutually exclusive, collectivelyexhaustive set of all possibleoutcomes.
Sor
i e lement
universal set
finite/discrete
infinite/continuous
20141-4
Probability Theory
Sample Space and Events
mutually exclusive
Exampleof two mutually exclusive events:One die is rolled. Event A is rolling a 1 or 2.Event B is rolling a 4 or 5.
Exampleof two non-mutually exclusive events:One die is rolled. Event A is rollinga 1 or 2. Event B is rolling a 2 or 3.
20141-5
Probability Theory
Sample Space and Events
collectively exhaustive
Example of collectively exhaustive events:One die is rolled. Event A is rolling anumber less than 5. Event B is rolling anumber greater than 3. Any roll of a die willsatisfy either A or B (in fact, a roll of 4satisfies both).
20141-6
Probability Theory
Sample Space and Events
mutually exclusive collect ively exhaustive
Example of events that are mutuallyexclusive and collectively exhaustive:You buy a stock today. The event A is thatthe price of the stock goes up tomorrow.The event B is the price of the stock goesdown tomorrow. Event C is the price ofthe stock does not change.
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20141-7
Probability Theory
Sample Space and Events
EventE
set of all outcomes (sample points)satisfying some property thatcharacterizes that event.se t
simple
compound
independent
dependent
Experiment
2014 1-8
Probability Theory
Sample Space and Events Example 1
Roll a normal six-sided die once
Each outcome is a number i = 1, , 6
6 distinct numbers: S = {1,2,3,4,5,6}Sample space
Sor
Outcomes
i
Events
E
E1 = set of all odd outcomes
E2 = set of all outcomes greater than 2
E3
= set of all outcomes that are thesquare of an integer
Experiment
2014 1-9
Probability Theory
Sample Space and Events Example 2
2 rolls of a four-sided die, recordboth numbers
Pairs of numbers {1,2,3,4} x {1,2,3,4}
16 distinct pairs if order matters;10 distinct pairs if order doesnt matter
Sample space
Sor
Outcomes
i
Events
E
E1
= set of all outcomes witha sum equal to 4
E2 = set of all outcomes with
an odd sum
Experiment
2014 1-10
Probability Theory
Sample Space and Events Example 3
2 rolls of a four-sided die, recordthe sum
Sum of the two numbers, a number fro2 and 8
{2,3,4,5,6,7,8}Sample space
Sor
Outcomes
i
Events
E
E1
= set of all even numbers{2, 4, 6, 8}
E2 = set of numbers > 5
{6, 7, 8}
2014 1-11
Probability Theory
Venn Diagrams: Flipping Three Coins
(a)True / FalseAll three Venn diagrams are the samplespace of the outcomes of flipping a cointhree times: {HHH, HHT, HTH, HTT, THH,THT, TTH, TTT}.
Sample Space and Events Example 4
TRUE
2014 1-12
Probability Theory
Venn Diagrams: Flipping Three Coins
(b) Event A could be described as: flipping a headno times / exactly two times
at least once / at most two times
Sample Space and Events Example 4
at least once
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2014 1-13
Probability Theory
Venn Diagrams: Flipping Three Coins
(c) Event B could be described as: flipping a headno times / exactly two times
at least once / at most two times
Sample Space and Events Example 4
at most two times
2014 1-14
Probability Theory
Venn Diagrams: Flipping Three Coins
(d) Event C could be described as: flipping a tailno times / exactly two times
at least once / at most two times
Sample Space and Events Example 4
at least once
2014 1-15
Probability Theory
Venn Diagrams: Flipping Three Coins
(d) Event D could be described as: flipping a headno times / exactly two times
at least once / at most two times
Sample Space and Events Example 4
exactly two times
2014 Probability Theory 1-16
Set Operations
Union = {x/ x in A or x in B}
Intersection = {x/ x in A and x in B}
Sometimes AB is written as A+BAB is written as AB
2014 Probability Theory 1-17
Complement = {x/ x in S and x not in A}
Difference = {x in A and x not in B}
= (A B = A Bc)
Set Operations
2014 Probability Theory 1-18
De Morgans Theorems
Set Operations
(1) (A B)c = Ac Bc
NOT in (A or B) = (NOT in A) AND (NOT in B)
(2) (A B)c = Ac Bc
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2014 Probability Theory 1-19
Event Operations
Let E1 = {a, b, c, d, e, f}E2 = {e, f, g, h}E3 = {i}
S = {a, b, c, d, e, f, g, h, i, j}
(a) E2c = {a, b, c, d, i, j}
(b) E1 E2 = {a, b, c, d, e, f, g, h}
(c) (E1 E2)E3 =
(d) (E1 E2 )c = {i, j}
2014 Probability Theory 1-20
Concept of Probability
Probability is a measure of the likelihood of anevent taking place once a randomexperiment is conducted.
2014 Probability Theory 1-21
1) 1 P[A] 0
Axioms of Probability
To every event A E, a real number P[A] thatsatisfies the following three axioms is assigned:
Probability is a nonnegative number.
2) P[S] = 1
S is the sure event.
2014 Probability Theory 1-22
Axioms of Probability
3) If A B = , then P [A B] = P [A] + P[B]
Additive property for disjoint events
2014 Probability Theory 1-23
Properties of Probability
i) P[] = 0
Properties derived from the three axioms:
The empty set (null set) is the impossible event.
2014 Probability Theory 1-24
Properties of Probability
ii) P [A] + P[Ac] = 1
Properties derived from the three axioms:
The empty set (null set) is the impossibleevent.
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2014 Probability Theory 1-25
Properties of Probability
iii) P[A B] = P[A] + P [B] P[A B]
Properties derived from the three axioms:
Additive property
2014 Probability Theory 1-26
Properties of Probability
iv) If A B, then P[B A] = P[B] P[A]and P[A] P[B]
Properties derived from the three axioms:
Finite Sample Spaces
Experiments which have finitely
many outcomes are said to havefinite sample spaces.
2014 1-27
Probability Theory
1 2 3 4 1 2 3
If we put these in a hat and pull one out at random, what is:
P(green square) =
Finite Sample Spaces
7
4
P(numbered 1) =7
2
20141-28
Probability Theory
1 2 3 4 1 2 3
If we put these in a hat and pull one out at random, what is:
P(NOT numbered 1) =
Finite Sample Spaces
7
5
7
5=
7
1
7
2+
7
4P(green OR numbered 1) =
20141-29
Probability Theory
Finite Sample Spaces Example
A die is loaded so that the numbers 2, 4 and6 are equally likely to appear. 1, 3, and 5are also equally likely to appear, but 1 isthree times more likely as 2 to appear.
(a) One die is rolled. Assign the probabilities tothe outcomes that accurately model thelikelihood of the various numbers to appear.
12
1=P(6)=P(4)=P(2)
12
3=P(5)=P(3)=P(1)
2014 1-30
Probability Theory
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Finite Probability Spaces Example
A die is loaded so that the numbers 2, 4 and6 are equally likely to appear. 1, 3, and 5are also equally likely to appear, but 1 isthree times more likely as 2 to appear.
(b) One die is rolled. What is the probability of getting a 5?
12
3=P(5)
2014 1-31
Probability Theory
Finite Probability Spaces Example
A die is loaded so that the numbers 2, 4 and
6 are equally likely to appear. 1, 3, and 5are also equally likely to appear, but 1 isthree times more likely as 2 to appear.
(c) One die is rolled. What is the probability of getting an even number?
12
3=P(6)+P(4)+P(2)
2014 1-32
Probability Theory
Finite Probability Spaces Example
A die is loaded so that the numbers 2, 4 and6 are equally likely to appear. 1, 3, and 5are also equally likely to appear, but 1 isthree times more likely as 2 to appear.
(d) One die is rolled. What is the probability of not getting a 5?
4
3=12
9=12
3-1
2014 1-33
Probability Theory
Finite Probability Spaces Example
A die is loaded so that the numbers 2, 4 and6 are equally likely to appear. 1, 3, and 5are also equally likely to appear, but 1 isthree times more likely as 2 to appear.
(e) Two dice are rolled. What is the probability ofgettingdoubles?
2
2
)12
3(=P(1)P(1)=1)P(1,
)12
1(=P(2)P(2)=2)P(2,
22 )12
33(+)
12
13(=P(doubles)
2014 1-34
Probability Theory
Finite Probability Spaces Example
A die is loaded so that the numbers 2, 4 and
6 are equally likely to appear. 1, 3, and 5are also equally likely to appear, but 1 isthree times more likely as 2 to appear.
(f) Two dice are rolled. What is the probability ofgetting a sum of 7?
P(1,6)+P(2,5)+P(3,4)+P(4,3)+P(5,2)+P(6,1)
)12
312
13(+)
12
112
33(=
)12
112
36(=
2014 1-35
Probability Theory
EquiprobableSpaces
If all the sample points within a given finiteprobability space are equal to each other,then it is known as an equiprobable space.
Examples:
A toss of a fair coin.
Select a name at random from a hat.
Throwing a well balanced die.
2014 1-36
Probability Theory
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EquiprobableSpaces
Theoretical Probability
=> finding the probability of eventsthat come from an equiprobablesample space.
n(S)
n(E)=)(EP
2014 1-37
Probability Theory
EquiprobableSpaces Example 1
A pair of fair dice is tossed. Determine the
probability that
(a) at least one of the dice shows a 6
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
0.30636
11
2014 1-38
Probability Theory
EquiprobableSpaces Example 1
A pair of fair dice is tossed. Determine theprobability that
(b) the sum of the two numbers is 5
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
0.1119
1=36
4
2014 1-39
Probability Theory
EquiprobableSpaces Example 2
From a group of 10 women and 5 men, 2people are selected at random to form acommittee. Find the probability that
(a) only men are selected
0.09521
2=
105
10
n(only men selected)= C(5, 2) = 10
n(2 person committees) = C(15, 2) = 105
2014 1-40
Probability Theory
EquiprobableSpaces Example 2
From a group of 10 women and 5 men, 2
people are selected at random to form acommittee. Find the probability of selecting
(b) exactly 1 man and 1 woman
0.47621
10=
105
50
n(exactly 1 man and 1 woman) = C(5, 1) = 50
n(2 person committees) = C(15, 2) = 105
2014 1-41
Probability Theory
EquiprobableSpaces Example 3
One student's name will be picked atrandom to win a CD player. There are 12male seniors, 15 female seniors, 10 male
juniors, 5 female juniors, 2 malesophomores, 4 female sophomores, 11 malefreshmen and 12 female freshman. Find theprobability that
(a) a senior or a junior is picked
0.59271
42=71
15+
71
27
2014 1-42
Probability Theory
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EquiprobableSpaces Example 3
One student's name will be picked atrandom to win a CD player. There are 12male seniors, 15 female seniors, 10 malejuniors, 5 female juniors, 2 malesophomores, 4 female sophomores, 11 malefreshmen and 12 female freshman. Find theprobability that
(b) a freshman or a female is picked
0.66271
47=
71
12-71
36+
71
23
2014 1-43
Probability Theory
EquiprobableSpaces Example 3
One student's name will be picked at
random to win a CD player. There are 12male seniors, 15 female seniors, 10 malejuniors, 5 female juniors, 2 malesophomores, 4 female sophomores, 11 malefreshmen and 12 female freshman. Find theprobability that
(c) a freshman is NOT picked
0.67671
48=
71
23-1=
P(E) = 1 - P(a freshman is picked)
2014 1-44
Probability Theory
EquiprobableSpaces Example 4
Lotto is a game where the player picks 6balls from a possible 49.
(a) What are the odds of getting the jackpot?(The player picks the same 6 balls as the 6chosen at the draw.)
To choose 6 numbers from 49 is C(49,6).Therefore, the odds are
816983,13,
1=
C(49,6)
1
2014 1-45
Probability Theory
EquiprobableSpaces Example 4
Lotto is a game where the player picks 6balls from a possible 49.
(b) What are the odds of getting just 4 balls?
possible winning 4-number combinations= C(6, 4) C(43, 2) = 15 903 = 13, 545
032,1
1=
816983,13,
54513,
2014 1-46
Probability Theory
Techniques of Counting
Mrs. Remalyn Q. Casem
email: [email protected]
2014 1-47Probability Theory
2014 Probability Theory 1-48
Fundamental Counting Principle and
Tree DiagramsPermutationsCircular PermutationsCombinations
TOPICS
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2014 1-49
Probability Theory
Fundamental Counting Principle
A small community consists of 10 women,each of whom has 3 children. If one womanand one of her children are to be chosen asmother and child of the year, how manydifferent choices are possible?
2014 1-50
Probability Theory
Fundamental Counting Principle
A college planning committee consists of 3freshmen, 4 sophomores, 5 juniors, and 2seniors. A subcommittee of 4, consisting of 1person from each class, is to be chosen. Howmany different subcommittees are possible?
2014 1-51
Probability Theory
Fundamental Counting Principle
How many different 7-place license platesare possible if the first 3 places are to beoccupied by letters and the final 4 bynumbers?
2014 1-52
Probability Theory
Fundamental Counting Principle
How many different 7-place license platesare possible if the first 3 places are to beoccupied by letters and the final 4 bynumbers if repetition among letters ornumbers were prohibited?
2014 1-53
Probability Theory
Tree Diagrams
Determine all the possible outcomes when a
coin is tossed three times.
2014 1-54
Probability Theory
Permutations
How many different ordered arrangements ofthe letters a, b,and c are possible?
abc, acb, bac, bca, cab, cba
Each arrangement is known as a permutation.
The different permutations of the n objects are
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2014 1-55
Probability Theory
Permutations
1. How many different battingorders are possible for a baseballteam consisting of 9 players?
The different permutations of the n objects are
2014 1-56
Probability Theory
PermutationsThe different permutations of the n objects are
2. A class in probability theory consists of6 men and 4 women. An examination isgiven, and the students are rankedaccording to their performance.Assume that no two students obtainthe same score.
(a) How many different rankings arepossible?
2014 1-57
Probability Theory
PermutationsThe different permutations of the n objects are
2. A class in probability theory consists of
6 men and 4 women. An examination isgiven, and the students are rankedaccording to their performance.Assume that no two students obtainthe same score.
(b) If the men are ranked justamong themselves and the women
just among them selves, how manydifferent rankings are possible?
2014 1-58
Probability Theory
PermutationsThe different permutations of the n objects are
3. Ms. Jones has 10 books that she isgoing to put on her bookshelf. Of these, 4 are mathematics books, 3 arechemistry books, 2 are history books,and 1 is a language book. Ms. Joneswants to arrange her books so that allthe books dealing with the samesubject are together on the shelf. Howmany different arrangements arepossible?
2014 1-59
Probability Theory
Permutations
We shall now determine the number of
permutations of a set of n objects whencertain of the objects are the same fromeach other. To set this situation straight inour minds, consider the following example.
2014 1-60
Probability Theory
Permutations
How many different letter arrangements can beformed from the letters PEPPER?
Hence, there are 6!/(3! 2!) = 60 possible letterarrangements of the letters PEPPER.
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2014 1-61
Probability Theory
Permutations
We shall now determine the number of permutations of a set of n objects whencertain of the objects are the same fromeach other. To set this situation straight inour minds, consider the following example.
different permutations of n objects,of which n1 are alike, n2 are alike,... , nr are alike.
2014 1-62
Probability Theory
Permutations
A chess tournament has 10 competitors, of
which 4 are Russian, 3 are from the UnitedStates, 2 are from Great Britain, and 1 isfrom Brazil. If the tournament result lists
just the nationalities of the players in theorder in which they placed, how manyoutcomes are possible?
How many different signals, each consistingof 9 flags hung in a line, can be made from aset of 4 white flags, 3 red flags, and 2 blueflags if all flags of the same color areidentical?
2014 1-63
Probability Theory
Combinations
To determine the number of different groupsof r objects that could be formed from atotal of n objects
A committee of 3 is to be formed from agroup of 20 people. How many differentcommittees are possible?
2014 1-64
Probability Theory
Combinations
From a group of 5 women and 7 men, howmany different committees consisting of 2women and 3 men can be formed?
What if 2 of the men are feuding and refuseto serve on the committee together?
EXERCISES
2014 1-65
Probability Theory
EXERCISES
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Probability Theory
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EXERCISES
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Probability Theory
EXERCISES
2014 1-68
Probability Theory
EXERCISES
2014 1-69
Probability Theory
EXERCISES
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Probability Theory
EXERCISES
2014 1-71
Probability Theory
EXERCISES
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Probability Theory
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EXERCISES
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Probability Theory
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Probability Theory
EXERCISES
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Probability Theory
EXERCISES
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Probability Theory
EXERCISES
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Probability Theory
EXERCISES
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Probability Theory
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EXERCISES
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Probability Theory
EXERCISES
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Probability Theory
EXERCISES
2014 1-81
Probability Theory
2014 Probability Theory 1-82
End of Presentation