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Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Math 1710 Class 4
Dr. Allen Back
Aug. 31, 2016
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Fancy Words
Random Phenomenon - Individually not predictable.But if repeated, long term relative frequency is.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Fancy Words
Random PhenomenonOutcome
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Fancy Words
Random Phenomenon -e.g. rolling a die
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Fancy Words
Random Phenomenon -e.g. rolling a dieOutcomee.g. a 3.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Fancy Words
Random Phenomenon -e.g. rolling a dieOutcomee.g. a 3.Sample Space S - Set of all possible outcomes.e.g. S = {1, 2, 3, 4, 5, 6}.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Fancy Words
Random Phenomenon -e.g. rolling a dieOutcomee.g. a 3.Sample Space S - Set of all possible outcomes.e.g. S = {1, 2, 3, 4, 5, 6}.Event E - a subset of the sample space – a set of outcomes.e.g. E = {2, 4, 6}“Rolling an even number.”
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Fancy Words
Random Phenomenon - Individually not predictable.But if repeated, long term relative frequency is.Random Phenomenon -e.g. tossing two coins
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Fancy Words
Random Phenomenon -e.g. tossing two coinsOutcomee.g. HT.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Fancy Words
Random Phenomenon -e.g. tossing two coinsOutcomee.g. HT.Sample Space S - Set of all possible outcomes.e.g. S = {HH,HT ,TH,TT}.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Fancy Words
Random Phenomenon -e.g. tossing two coinsOutcomee.g. HT.Sample Space S - Set of all possible outcomes.e.g. S = {HH,HT ,TH,TT}.Event E - a subset of the sample space – a set of outcomes.e.g. E = {HT ,TH}“Rolling a total of 1 head.”
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Probability Assignment on a Sample Space S
We won’t usually be this formal!(1) 0 ≤ P(Outcome) ≤ 1.(2) Σ P(All Outcomes in S)= 1.(3) Σ P(Outcomes in an Event)=P(Event).
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Probability Assignment on a Sample Space S
We won’t usually be this formal!(1) 0 ≤ P(Outcome) ≤ 1.(2) Σ P(All Outcomes in S)= 1.(3) Σ P(Outcomes in an Event)=P(Event).
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Probability Assignment on a Sample Space S
We won’t usually be this formal!(1) 0 ≤ P(Outcome) ≤ 1.(2) Σ P(All Outcomes in S)= 1.(3) Σ P(Outcomes in an Event)=P(Event).
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Probability Properties
(1) 0 ≤ P(E ) ≤ 1 for any event E .(2) P(φ) = 0, P(S) = 1.(3) P(A ∪ B) = P(A) + P(B) if A and B are disjoint.(4) P(Ac) = 1− P(A)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Probability Properties
(1) 0 ≤ P(E ) ≤ 1 for any event E .(2) P(φ) = 0, P(S) = 1.(3) P(A ∪ B) = P(A) + P(B) if A and B are disjoint.(4) P(Ac) = 1− P(A)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Probability Properties
(1) 0 ≤ P(E ) ≤ 1 for any event E .(2) P(φ) = 0, P(S) = 1.(3) P(A ∪ B) = P(A) + P(B) if A and B are disjoint.(4) P(Ac) = 1− P(A)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Probability Properties
(1) 0 ≤ P(E ) ≤ 1 for any event E .(2) P(φ) = 0, P(S) = 1.(3) P(A ∪ B) = P(A) + P(B) if A and B are disjoint.(4) P(Ac) = 1− P(A)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Probability Assignment
English language frequencies:E 13.0% H 3.5% W 1.6T 9.3% L 3.5% V 1.3%N 7.8% C 3.0% B 0.9%R 7.7% F 2.8% X 0.5%O 7.4% P 2.7% K 0.3%I 7.4% U 2.7% Q 0.3%A 7.3% M 2.5% J 0.2%S 6.3% Y 1.9% Z 0.1%D 4.4% G 1.6%
Sample Space S: S={A,B,C,...Z}.An Outcome: e.g. an M.An Event E: e.g. being a vowel. Formally: E={A,E,I,O,U}P(E ) = 13.0 + 7.4 + 7.4 + 7.3 + 2.7% = 37.8%.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Probability Assignment
English language frequencies:E 13.0% H 3.5% W 1.6T 9.3% L 3.5% V 1.3%N 7.8% C 3.0% B 0.9%R 7.7% F 2.8% X 0.5%O 7.4% P 2.7% K 0.3%I 7.4% U 2.7% Q 0.3%A 7.3% M 2.5% J 0.2%S 6.3% Y 1.9% Z 0.1%D 4.4% G 1.6%
Sample Space S: S={A,B,C,...Z}.
An Outcome: e.g. an M.An Event E: e.g. being a vowel. Formally: E={A,E,I,O,U}P(E ) = 13.0 + 7.4 + 7.4 + 7.3 + 2.7% = 37.8%.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Probability Assignment
English language frequencies:E 13.0% H 3.5% W 1.6T 9.3% L 3.5% V 1.3%N 7.8% C 3.0% B 0.9%R 7.7% F 2.8% X 0.5%O 7.4% P 2.7% K 0.3%I 7.4% U 2.7% Q 0.3%A 7.3% M 2.5% J 0.2%S 6.3% Y 1.9% Z 0.1%D 4.4% G 1.6%
Sample Space S: S={A,B,C,...Z}.An Outcome: e.g. an M.
An Event E: e.g. being a vowel. Formally: E={A,E,I,O,U}P(E ) = 13.0 + 7.4 + 7.4 + 7.3 + 2.7% = 37.8%.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Probability Assignment
English language frequencies:E 13.0% H 3.5% W 1.6T 9.3% L 3.5% V 1.3%N 7.8% C 3.0% B 0.9%R 7.7% F 2.8% X 0.5%O 7.4% P 2.7% K 0.3%I 7.4% U 2.7% Q 0.3%A 7.3% M 2.5% J 0.2%S 6.3% Y 1.9% Z 0.1%D 4.4% G 1.6%
Sample Space S: S={A,B,C,...Z}.An Outcome: e.g. an M.An Event E: e.g. being a vowel. Formally: E={A,E,I,O,U}P(E ) = 13.0 + 7.4 + 7.4 + 7.3 + 2.7% = 37.8%.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Prob of Event frm Prob Outcomes
Sample Space S: S={A,B,C,...Z}.An Outcome: e.g. an M.An Event E: e.g. being a vowel. Formally: E={A,E,I,O,U}P(E ) = 13.0 + 7.4 + 7.4 + 7.3 + 2.7% = 37.8%.
E2: a letter in ‘HELLO”.E2 = {H,E , L,O}.P(E2) = 3.5 + 13 + 3.5 + 7.4% = 27.4%.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Prob of Event frm Prob Outcomes
Sample Space S: S={A,B,C,...Z}.An Outcome: e.g. an M.An Event E: e.g. being a vowel. Formally: E={A,E,I,O,U}P(E ) = 13.0 + 7.4 + 7.4 + 7.3 + 2.7% = 37.8%.E2: a letter in ‘HELLO”.
E2 = {H,E , L,O}.P(E2) = 3.5 + 13 + 3.5 + 7.4% = 27.4%.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Prob of Event frm Prob Outcomes
Sample Space S: S={A,B,C,...Z}.An Outcome: e.g. an M.An Event E: e.g. being a vowel. Formally: E={A,E,I,O,U}P(E ) = 13.0 + 7.4 + 7.4 + 7.3 + 2.7% = 37.8%.E2: a letter in ‘HELLO”.E2 = {H,E , L,O}.
P(E2) = 3.5 + 13 + 3.5 + 7.4% = 27.4%.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Prob of Event frm Prob Outcomes
Sample Space S: S={A,B,C,...Z}.An Outcome: e.g. an M.An Event E: e.g. being a vowel. Formally: E={A,E,I,O,U}P(E ) = 13.0 + 7.4 + 7.4 + 7.3 + 2.7% = 37.8%.E2: a letter in ‘HELLO”.E2 = {H,E , L,O}.P(E2) = 3.5 + 13 + 3.5 + 7.4% = 27.4%.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Addition Rule
E2 = {H,E , L.O}.P(E2) = 3.5 + 13 + 3.5 + 7.4% = 27.4%.E3 = {B,A,D}.P(E3) = 0.9 + 7.3 + 4.4% = 12.6%.E2 and E3 are disjoint events.
P(E2 ∪ E3) = P(E2) + P(E3) = 27.4 + 12.6%.E2 ∪ E3 means “a letter in HELLO or BAD.”E2 ∪ E3 = {H,E , L,O,B,A,D}.Union (∪) corresponds to “or’.’
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Addition Rule
E2 = {H,E , L.O}.P(E2) = 3.5 + 13 + 3.5 + 7.4% = 27.4%.E3 = {B,A,D}.P(E3) = 0.9 + 7.3 + 4.4% = 12.6%.E2 and E3 are disjoint events.P(E2 ∪ E3) = P(E2) + P(E3) = 27.4 + 12.6%.
E2 ∪ E3 means “a letter in HELLO or BAD.”E2 ∪ E3 = {H,E , L,O,B,A,D}.Union (∪) corresponds to “or’.’
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Addition Rule
E2 = {H,E , L.O}.P(E2) = 3.5 + 13 + 3.5 + 7.4% = 27.4%.E3 = {B,A,D}.P(E3) = 0.9 + 7.3 + 4.4% = 12.6%.E2 and E3 are disjoint events.P(E2 ∪ E3) = P(E2) + P(E3) = 27.4 + 12.6%.E2 ∪ E3 means “a letter in HELLO or BAD.”
E2 ∪ E3 = {H,E , L,O,B,A,D}.Union (∪) corresponds to “or’.’
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Addition Rule
E2 = {H,E , L.O}.P(E2) = 3.5 + 13 + 3.5 + 7.4% = 27.4%.E3 = {B,A,D}.P(E3) = 0.9 + 7.3 + 4.4% = 12.6%.E2 and E3 are disjoint events.P(E2 ∪ E3) = P(E2) + P(E3) = 27.4 + 12.6%.E2 ∪ E3 means “a letter in HELLO or BAD.”E2 ∪ E3 = {H,E , L,O,B,A,D}.
Union (∪) corresponds to “or’.’
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Addition Rule
E2 = {H,E , L.O}.P(E2) = 3.5 + 13 + 3.5 + 7.4% = 27.4%.E3 = {B,A,D}.P(E3) = 0.9 + 7.3 + 4.4% = 12.6%.E2 and E3 are disjoint events.P(E2 ∪ E3) = P(E2) + P(E3) = 27.4 + 12.6%.E2 ∪ E3 means “a letter in HELLO or BAD.”E2 ∪ E3 = {H,E , L,O,B,A,D}.Union (∪) corresponds to “or’.’
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Equally Likely Outcomes
Sometimes all n outcomes in a sample space are equally likely.
e.g. fair die, fair coins, . . .If so, each has probability 1
n .So the sample space S = {HH,HT ,TH,TT} for tossing 2coins shows us
Num Heads Prob.0 1
41 1
22 1
4
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Equally Likely Outcomes
Sometimes all n outcomes in a sample space are equally likely.e.g. fair die, fair coins, . . .
If so, each has probability 1n .
So the sample space S = {HH,HT ,TH,TT} for tossing 2coins shows us
Num Heads Prob.0 1
41 1
22 1
4
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Equally Likely Outcomes
Sometimes all n outcomes in a sample space are equally likely.e.g. fair die, fair coins, . . .If so, each has probability 1
n .
So the sample space S = {HH,HT ,TH,TT} for tossing 2coins shows us
Num Heads Prob.0 1
41 1
22 1
4
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Equally Likely Outcomes
Sometimes all n outcomes in a sample space are equally likely.e.g. fair die, fair coins, . . .If so, each has probability 1
n .So the sample space S = {HH,HT ,TH,TT} for tossing 2coins shows us
Num Heads Prob.0 1
41 1
22 1
4
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence Informally
Independence of A and B means knowing A happened doesn’taffect the chance of B happening.
Product Rule for independent events: P(A ∩ B) = P(A)P(B).Intersection (∩) corresponds to “and’.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence Informally
Independence of A and B means knowing A happened doesn’taffect the chance of B happening.Product Rule for independent events: P(A ∩ B) = P(A)P(B).
Intersection (∩) corresponds to “and’.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence Informally
Independence of A and B means knowing A happened doesn’taffect the chance of B happening.Product Rule for independent events: P(A ∩ B) = P(A)P(B).
Intersection (∩) corresponds to “and’.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence Informally
Independence of A and B means knowing A happened doesn’taffect the chance of B happening.Product Rule for independent events: P(A ∩ B) = P(A)P(B).Intersection (∩) corresponds to “and’.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Customers Spin a Wheel
12 equal size areas.Wheel gives customer discount in %.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08
(1) Prob of at least a 40% discount?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08
(2) Prob two customers in a row get 10% discount?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08
(3) Prob 3 consecutive customers get 20%?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08
(4) Prob none of first four get over 20%?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08
(5) Prob that first 100% is fifth customer?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08
(6) Chance that at least one of the first six shoppers gets lessthan a 100% discount?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08(1) Prob of at least a 40% discount?
.17 + .08 = .25(2) Prob two customers in a row get 10% discount?
.5 .5.52 = .25.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08(1) Prob of at least a 40% discount?.17 + .08 = .25
(2) Prob two customers in a row get 10% discount?
.5 .5.52 = .25.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08(1) Prob of at least a 40% discount?.17 + .08 = .25(2) Prob two customers in a row get 10% discount?
.5 .5.52 = .25.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08(1) Prob of at least a 40% discount?.17 + .08 = .25(2) Prob two customers in a row get 10% discount?
.5 .5
.52 = .25.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Spinner Prob Asst
Discount Probability10% 6/12=.520% 3/12=.2540% 2/12=.17
100% 1/12=.08(1) Prob of at least a 40% discount?.17 + .08 = .25(2) Prob two customers in a row get 10% discount?
.5 .5.52 = .25.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Venn Diagram Picture
(No longer assuming A and B are disjoint events.)
31 2
A = {1, 2} B = {2, 3}A ∩ B = {2} A ∪ B = {1, 2, 3}
P(A ∪ B) =P(A) + P(B)− P(A ∩ B)
1, 2, 3 1, 2 2, 3 2
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Playing Sports
Suppose 30% play soccer, 20% basketball, 12% both.Probability of playing at least one?
Soln: .30 + .20− .12 = .38.or
.62
S B.12.18 .08
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Playing Sports
Suppose 30% play soccer, 20% basketball, 12% both.Probability of playing at least one?Soln: .30 + .20− .12 = .38.
or
.62
S B.12.18 .08
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Playing Sports
Suppose 30% play soccer, 20% basketball, 12% both.Probability of playing at least one?Soln: .30 + .20− .12 = .38.or
.62
S B.12.18 .08
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Ace or Red
One card is drawn.Probability of an ace or red card?
Direct counting possible:26 red cards + 2 black aces out of 52.Inclusion/Exclusion:P(Ace or Red) = P(Ace) +P(Red) - P(Ace and Red)
=4
52+
26
52− 2
52=
28
52
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Ace or Red
One card is drawn.Probability of an ace or red card?Direct counting possible:26 red cards + 2 black aces out of 52.
Inclusion/Exclusion:P(Ace or Red) = P(Ace) +P(Red) - P(Ace and Red)
=4
52+
26
52− 2
52=
28
52
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Ace or Red
One card is drawn.Probability of an ace or red card?Direct counting possible:26 red cards + 2 black aces out of 52.Inclusion/Exclusion:P(Ace or Red) = P(Ace) +P(Red) - P(Ace and Red)
=4
52+
26
52− 2
52=
28
52
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Smallest Card a Queen
Two cards dealt.Prob(smallest a queen)?(Q viewed as smallest when two Q’s.Ace viewed as high.)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Smallest Card a Queen
Two cards dealt.Prob(smallest a queen)?
Approach:
A = event 1st card =Q & 2nd a Q,K or A.
B = event 2nd card =Q & 1st a Q,K or A.
P(A or B) wanted!
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Smallest Card a Queen
Two cards dealt.Prob(smallest a queen)?
Approach:
A = event 1st card =Q & 2nd a Q,K or A.
B = event 2nd card =Q & 1st a Q,K or A.
P(A or B)=P(A)+P(B)-P(A and B)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Smallest Card a Queen
Two cards dealt.Prob(smallest a queen)?
A = event 1st card =Q & 2nd a Q,K or A.
P(A) =4
52· 11
51=
44
51 · 52
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Smallest Card a Queen
Two cards dealt.Prob(smallest a queen)?
B = event 2nd card =Q & 1st a Q,K or A.
Q
event B
>=Q
4/5211/51
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Smallest Card a Queen
Two cards dealt.Prob(smallest a queen)?
B = event 2nd card =Q & 1st a Q,K or A.
Q
event B
>=Q
4/5211/51P(B) =
44
51 · 52too.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Smallest Card a Queen
Two cards dealt.Prob(smallest a queen)?
P(A and B) =4
52· 3
51=
12
51 · 52
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Smallest Card a Queen
Two cards dealt.Prob(smallest a queen)?
P(A or B)=P(A)+P(B)-P(A and B)
So P(A or B) =44
51 · 52+
44
51 · 52− 12
51 · 52=
76
51 · 52
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Definition
”Probability of A given B.”
P(A|B) =P(A ∩ B)
P(B)
31 2
A = {1, 2} B = {2, 3}A ∩ B = {2}
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Definition
”Probability of A given B.”
P(A|B) =P(A ∩ B)
P(B)
31 2
A = {1, 2} B = {2, 3}A ∩ B = {2}
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Life Tables
Really a table of probabilities;e.g. P(woman lives to 45)=.963.Will write as P(45).
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Life Tables
P(80|60)?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Life Tables
P(80|60) =P(80 ∩ 60)
P(60)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Life Tables
P(80|60) =P(80 ∩ 60)
P(60)=
P(80)
P(60)=.571
.898= .636
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Life Tables
P(80|60) =P(80 ∩ 60)
P(60)=
P(80)
P(60)=.571
.898= .636
P(60|80) =P(80 ∩ 60)
P(80)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Life Tables
P(80|60) =P(80 ∩ 60)
P(60)=
P(80)
P(60)=.571
.898= .636
P(60|80) =P(80 ∩ 60)
P(80)=
P(80)
P(80)= 1
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Life Tables
P(80|60) =P(80 ∩ 60)
P(60)=
P(80)
P(60)=.571
.898= .636
P(60|80) =P(80 ∩ 60)
P(80)=
P(80)
P(80)= 1
P(60|80) 6= P(80|60).
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence
A and B are independent events if
P(A|B) = P(A)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence
A and B are independent events if
P(A|B) = P(A)
Suppose 10% of all cars have sticky valves10% have oil leaks1% both.Are sticky valves and oil leaks independent?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence
A and B are independent events if
P(A|B) = P(A)
Suppose 10% of all cars have sticky valves10% have oil leaks1% both.Are sticky valves and oil leaks independent?
P(sticky |leaky) =P(both)
P(leaky)=.01
.10= .10 = P(sticky).
So they are independent.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence
A and B are independent events if
P(A|B) = P(A)
Box has 2 white tickets labeled 12 black labeled 11 white labeled 81 black one labeled 6.Are drawing a 1 and color black independent?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence
A and B are independent events if
P(A|B) = P(A)
Box has 2 white tickets labeled 12 black labeled 11 white labeled 81 black one labeled 6.Are drawing a 1 and color black independent?
P(1|black) =2
3P(1) =
4
6Equal, so they are independent.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence
A and B are independent events if
P(A|B) = P(A)
Box has 2 white tickets labeled 12 black labeled 11 white labeled 81 black one labeled 6.Are drawing an 8 and color black independent?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Independence
A and B are independent events if
P(A|B) = P(A)
Box has 2 white tickets labeled 12 black labeled 11 white labeled 81 black one labeled 6.Are drawing an 8 and color black independent?
P(8|black) =0
3P(8) =
1
6Not equal, so not independent.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Collins Case
Purse Snatching Suspects:
Yellow Auto: 1/10Man with Mustache: 1/4Woman with Ponytail: 1/10Woman with Blond Hair: 1/3Black Man with Beard: 1/10Interracial Couple in Car: 1/1000
Math Prof in LA:1/12,000,000 chance any couple matches these.
Defendants convicted.
Reversed on appeal.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Collins Case
Purse Snatching Suspects:
Yellow Auto: 1/10Man with Mustache: 1/4Woman with Ponytail: 1/10Woman with Blond Hair: 1/3Black Man with Beard: 1/10Interracial Couple in Car: 1/1000
Math Prof in LA:1/12,000,000 chance any couple matches these.
Defendants convicted.
Reversed on appeal.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Collins Case
Purse Snatching Suspects:
Yellow Auto: 1/10Man with Mustache: 1/4Woman with Ponytail: 1/10Woman with Blond Hair: 1/3Black Man with Beard: 1/10Interracial Couple in Car: 1/1000
Math Prof in LA:1/12,000,000 chance any couple matches these.
Defendants convicted.
Reversed on appeal.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Collins Case
Purse Snatching Suspects:
Yellow Auto: 1/10Man with Mustache: 1/4Woman with Ponytail: 1/10Woman with Blond Hair: 1/3Black Man with Beard: 1/10Interracial Couple in Car: 1/1000
Math Prof in LA:1/12,000,000 chance any couple matches these.
Defendants convicted.
Reversed on appeal.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Prosecutor’s Fallacy
Just because
P(DNA evidence | innocent)
is small, it doesn’t follow that
P(innocent | DNA evidence)
is small.
“The chance that anyone else but the appellant left the hairsat the scene of the crime is 6 billion to 1.” (1991)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Prosecutor’s Fallacy
Just because
P(DNA evidence | innocent)
is small, it doesn’t follow that
P(innocent | DNA evidence)
is small.
“The chance that anyone else but the appellant left the hairsat the scene of the crime is 6 billion to 1.” (1991)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Defendant’s Fallacy
1 in a million match this DNA
So there are hundreds of matchesand only a small chance the defendant is guilty.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Defendant’s Fallacy
1 in a million match this DNA
So there are hundreds of matchesand only a small chance the defendant is guilty.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
Jeans OtherMale 12 5
Female 8 11
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
P(M) ?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
P(M) ?
P(M) =17
36
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
P(Male wears jeans)?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
P(Male wears jeans)?i.e. P(J|M)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
P(Male wears jeans)?i.e. P(J|M)
P(J|M) =P(J ∩M)
P(M)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
P(Male wears jeans)?i.e. P(J|M)
P(J|M) =P(J ∩M)
P(M)=
12
17
More precisely:12
36/
17
36
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
P(Someone wearing jeans is male)?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
P(Someone wearing jeans is male)?i.e. P(M|J)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
P(Someone wearing jeans is male)?i.e. P(M|J)
P(M|J) =P(J ∩M)
P(J)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
P(Someone wearing jeans is male)?i.e. P(M|J)
P(M|J) =P(J ∩M)
P(J)=
12
20
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
Are gender and wearing jeans disjoint?independent?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
Are gender and wearing jeans disjoint?independent?
P(M ∩ J) > 0
So they sometimes both happen.Not disjoint.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
From a 2 way table
With Marginal Distributions:Jeans Other
Male 12 5 17Female 8 11 19
20 16 36
Are gender and wearing jeans disjoint?independent?
P(M|J) 6= P(M)
So not independent.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Most Pairs of Events are Neither!
Checking disjointness doesn’t even require probabilities.Checking independence does.But when events are related in one of these ways, it is worthnoting.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Most Pairs of Events are Neither!
Checking disjointness doesn’t even require probabilities.Checking independence does.But when events are related in one of these ways, it is worthnoting.Sps:A=getting swine fluB=getting a shot.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Most Pairs of Events are Neither!
Checking disjointness doesn’t even require probabilities.Checking independence does.But when events are related in one of these ways, it is worthnoting.Sps:A=getting swine fluB=getting a shot.What would disjoint/independent mean here?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Disjointness and Independence Never HappenTogether (almost)
This holds as long as none of the events have probability 0.
For if A and B are disjoint,
P(A|B) =P(A ∩ B)
P(B)= 0 6= P(A).
So A and B are not independent.
(Events of probability 0 do sometimes happen; e.g. a personexactly 173 cm. tall.)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Disjointness and Independence Never HappenTogether (almost)
This holds as long as none of the events have probability 0.
For if A and B are disjoint,
P(A|B) =P(A ∩ B)
P(B)= 0 6= P(A).
So A and B are not independent.
(Events of probability 0 do sometimes happen; e.g. a personexactly 173 cm. tall.)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Disjointness and Independence Never HappenTogether (almost)
This holds as long as none of the events have probability 0.
For if A and B are disjoint,
P(A|B) =P(A ∩ B)
P(B)= 0 6= P(A).
So A and B are not independent.
(Events of probability 0 do sometimes happen; e.g. a personexactly 173 cm. tall.)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Disjointness and Independence Never HappenTogether (almost)
This holds as long as none of the events have probability 0.
For if A and B are disjoint,
P(A|B) =P(A ∩ B)
P(B)= 0 6= P(A).
So A and B are not independent.
(Events of probability 0 do sometimes happen; e.g. a personexactly 173 cm. tall.)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Simple Sequential Example
Two games against same opponent.P(win 1st)=.4If you win 1st, P(win 2nd)=.2If you lose 1st, P(win 2nd)=.3P(win both)=?; P(win exactly 1 of 2) = ?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Sequences of events
Tree diagrams are good for keeping track of all thecombinations in multi-stage events.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Sequences of events
Tree diagrams are good for keeping track of all thecombinations in multi-stage events.Graphical version of the mult rule P(A ∩ B) = P(B|A)P(A).
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Sequences of events
Tree diagrams are good for keeping track of all thecombinations in multi-stage events.Graphical version of the mult rule P(A ∩ B) = P(B|A)P(A).First Step:
notA
A
A
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Sequences of events
Tree diagrams are good for keeping track of all thecombinations in multi-stage events.Graphical version of the mult rule P(A ∩ B) = P(B|A)P(A).Second Step:
notA
A
notB
notBB
A B
B
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Sequences of events
Tree diagrams are good for keeping track of all thecombinations in multi-stage events.Graphical version of the mult rule P(A ∩ B) = P(B|A)P(A).Full First Step:
P(not A)
P(A)
A
A
not A
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Sequences of events
Tree diagrams are good for keeping track of all thecombinations in multi-stage events.Graphical version of the mult rule P(A ∩ B) = P(B|A)P(A).Full Second Step:
P(not A)
A
P(B|A)
B
P(notB|A)
P(B|not A)
P(notB|notA)
P(A)A
notA
not BB
not B
B
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
Sequences of events
Tree diagrams are good for keeping track of all thecombinations in multi-stage events.Graphical version of the mult rule P(A ∩ B) = P(B|A)P(A).Final After Multiplying:
P(not A)
A
P(B|A)
B
P(notB|A)
P(B|not A)
P(notB|notA)
P(A and B)
P(A and not B)
P(not B and not A)
P(A)A
notA
not BB
not B
B
P(B and not A)
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Simple Sequential Example
Two games against same opponent.P(win 1st)=.4If you win 1st, P(win 2nd)=.2If you lose 1st, P(win 2nd)=.3P(win both)=?; P(win exactly 1 of 2) = ?
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Simple Sequential Example
Two games against same opponent.P(win 1st)=.4If you win 1st, P(win 2nd)=.2If you lose 1st, P(win 2nd)=.3P(win both)=?; P(win exactly 1 of 2) = ?Solution: A=win 1st, B=win 2nd
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Simple Sequential Example
Two games against same opponent.P(win 1st)=.4If you win 1st, P(win 2nd)=.2If you lose 1st, P(win 2nd)=.3P(win both)=?; P(win exactly 1 of 2) = ?First Step:
wn1
ls1
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Simple Sequential Example
Two games against same opponent.P(win 1st)=.4If you win 1st, P(win 2nd)=.2If you lose 1st, P(win 2nd)=.3P(win both)=?; P(win exactly 1 of 2) = ?Second Step:
wn1
ls1
wn2
ls2
ls2
1st game 2nd game
wn2
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Simple Sequential Example
Two games against same opponent.P(win 1st)=.4If you win 1st, P(win 2nd)=.2If you lose 1st, P(win 2nd)=.3P(win both)=?; P(win exactly 1 of 2) = ?Probabilities:
wn1
ls1
wn2
ls2
ls2
1st game 2nd game
wn2
.4
.6
.2
.8
.3
.7
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Simple Sequential Example
Two games against same opponent.P(win 1st)=.4If you win 1st, P(win 2nd)=.2If you lose 1st, P(win 2nd)=.3P(win both)=?; P(win exactly 1 of 2) = ?Probabilities:
wn1
ls1
wn2
ls2
ls2
1st game 2nd game
wn2
.4
.6
.2
.8
.3
.7
(.4)(.2)=.08
(.4)(.8)=.32
(.6)(.3)=.18
(.6)(.7)=.42
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Simple Sequential Example
Two games against same opponent.P(win 1st)=.4If you win 1st, P(win 2nd)=.2If you lose 1st, P(win 2nd)=.3P(win both)=?; P(win exactly 1 of 2) = ?Probabilities:
wn1
ls1
wn2
ls2
ls2
1st game 2nd game
wn2
.4
.6
.2
.8
.3
.7
(.4)(.2)=.08
(.4)(.8)=.32
(.6)(.3)=.18
(.6)(.7)=.42
Note: .08+.32+.18+.42=1since they count allthe disjoint possibilitiesof 1st game/2nd game.
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Simple Sequential Example
Two games against same opponent.P(win 1st)=.4If you win 1st, P(win 2nd)=.2If you lose 1st, P(win 2nd)=.3P(win both)=?; P(win exactly 1 of 2) = ?Probabilities:
wn1
ls1
wn2
ls2
ls2
1st game 2nd game
wn2
.4
.6
.2
.8
.3
.7
(.4)(.2)=.08
(.4)(.8)=.32
(.6)(.3)=.18
(.6)(.7)=.42
Note: .08+.32+.18+.42=1since they count allthe disjoint possibilitiesof 1st game/2nd game.
P(wn 1 of 2)=.32+.18=.50
P(wn 2nd) = .08+.18=.26
Math 1710Class 4
V1
FormalProbability
Letters
IndependentEvents
Spinner
Gen. AdditionRule
Smallest ofTwo a Queen
ConditionalProbability
Independence
Misapplications
Cnd. Prb. &Tables
Disjoint vs.Independent
BeginningTree Diagrams
A Simple Sequential Example
Two games against same opponent.P(win 1st)=.4If you win 1st, P(win 2nd)=.2If you lose 1st, P(win 2nd)=.3P(win both)=?; P(win exactly 1 of 2) = ?Probabilities:
wn1
ls1
wn2
ls2
ls2
1st game 2nd game
wn2
.4
.6
.2
.8
.3
.7
(.4)(.2)=.08
(.4)(.8)=.32
(.6)(.3)=.18
(.6)(.7)=.42
Note: .08+.32+.18+.42=1since they count allthe disjoint possibilitiesof 1st game/2nd game.
P(wn 1 of 2)=.32+.18=.50
P(wn 2nd) = .08+.18=.26
P(wn 1st|wn 2nd)= P(wn1st&wn2nd)/P(wn2nd) =.08/.26=.308
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