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**Unformatted text preview: **Math012 Basic Statistics Ping-Shi Wu Department of Mathematics Lehigh University September 15,17,19 2008 Ping-Shi Wu (Lehigh University) M ath012 September 15,17,19 2008 1 / 23 What is Conditional Probability? 1 Recall that ordinary probability, P ( A ), delineates the relative likelihood of an event, A, against the whole sample space ( S ). The Conditional Probability, P ( A | B ), on the other hand describes the relative likelihood of event A against a SUBSET in sample space, B . For example in tossing a coin twice, A refers to ”getting Head on the second toss”, B refers to ”getting at least one Head”, P ( A ) = # { HH , TH } # { HH , HT , TH , TT } = 2 4 = 1 2 , P ( A | B ) = # { HH , TH } # { HH , HT , TH } = 2 3 . 2 Are the following two statements the same? In US, more than half of people drink Budweisier when they drink beer in the bar. In US, more than half of people drink Budweisier. If you were the Bud owner, which situation will you be happier? Ping-Shi Wu (Lehigh University) M ath012 September 15,17,19 2008 2 / 23 § 4.5 Conditional Probability and Independence Given that both parents having Blood Type A and the first child having Blood Type O, what is the probability that the second child will have Blood Type A? A bit biology: AA:pure A, Ai: hybrid A, both yield A; ii: O. Given that both parents have A and the first child having O, it means both parents are hybrid A. A i A AA Ai i iA ii Under the given condition, the new sample space is { AA, Ai, iA, ii } and P ( A ) = P ( AA ) + P ( Ai ) + P ( iA ) = 3 4 . Ping-Shi Wu (Lehigh University) M ath012 September 15,17,19 2008 3 / 23 § 4.5 Conditional Probability and Independence A: Admitted. B: Male. Admitted Not Admitted Total Male 233(.418) 324(.582) 557 Female 88(.312) 194(.688) 282 Total 321 518 839 P ( A | B ) = 233 557 = . 418. P ( ¯ A | B ) = 324 557 = . 582 = 1- . 418 = 1- P ( A | B ). P ( A ) = 321 839 ; P ( B ) = 557 839 P ( AB ) = 233 839 P ( A | B ) = P ( AB ) P ( B ) = 233 / 839 557 / 839 = 0 . 418 Not a coincident, P ( A | B ) = P ( AB ) P ( B ) is the definition for conditional probability of A given B. Ping-Shi Wu (Lehigh University) M ath012 September 15,17,19 2008 4 / 23 § 4.5 Conditional Probability and Independence Events A and B are called independent events if one of the following is true: 1 P ( A | B ) = P ( A ). 2 P ( B | A ) = P ( B ). 3 P ( AB ) = P ( A ) P ( B ). Recall the coin-flipping example (flipping a coin twice): A: First flip is a H. B: Second flip is a H. P ( A | B ) = P ( A ), so events A and B are independent. Also you might say since P ( AB ) = 1 4 = 1 2 · 1 2 = P ( A ) · P ( B ). So A and B are independent....

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