1.4 lecture04

1.4 lecture04 - Monty Hall: smarter answer STAT 1301...

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1 STAT 1301 Lecture 4 Monty Hall: smarter answer A B C, the key is in A If I choose A, then switch I always lose If I choose B or C then switch I always If I choose B or C, then switch I always win P(A)=1/3, P(B or C)=2/3 What if A B C D, the key is in A, switch or not? Poker Game 2, 3, 4, 5, 6, 7, 8, 9, 10 2 7 9 3 5 10 4 6 8 • Probabilities are not transitive. Multiplication Rule Conditional probability: P( A | B ) = P( A B ) / P( B ) Multiplication Rule: For any two events A and B and , P( A B ) = P( A ) P(B|A) More generally, P( A 1 A 2 ··· A n ) = P( A 1 ) P( A 2 | A 1 ) ··· P( A n | A 1 …A n-1 ) Example A family has two children. Assume all four possible “outcomes” -- (younger is a boy, older is a girl), …, are equally likely. What is the probability that both are boys given that at least one is a boy? BG BB GG GB BG BB GG GB Let A = Both children are boys, B = At least one is a boy. Then P( A ) = 1/4, P( B ) = 3/4. (Why?) P(B c )=P(both are girls) = 1/4. P(B)= 1- P(B c ) = 3/4. So, P( A | B )=P( A B )/P( B ) = P( A )/P( B ) = 1/3.
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2 Example An urn contains 6 white balls and 5 blue balls. Two balls are drawn sequentially without
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1.4 lecture04 - Monty Hall: smarter answer STAT 1301...

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