# lect5 - THE LAW OF CASES AND OF SUBCASES EXCHANGEABILITY...

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THE LAW OF CASES, AND OF SUBCASES Outline THE LAW OF CASES, AND OF SUBCASES EXCHANGEABILITY 1 / 13 Xinghua Zheng Lect 5: Law of Cases/Subcases

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THE LAW OF CASES, AND OF SUBCASES THE LAW OF CASES, AND OF SUBCASES Suppose you want to ﬁnd the conditional chance P [ A | C ] for an event A to occur, in the case that an event C has occurred. Suppose C can be partitioned into subcases C 1 , C 2 ,... , i.e., the C i ’s are mutually exclusive and their union is C . The events A C 1 , A C 2 are and their union is . Thus P [ A | C ] = P [ A C ] P [ C ] = X i P [ C i ] P [ C i ] P [ C ] = X i . 2 / 13 Xinghua Zheng Lect 5: Law of Cases/Subcases
THE LAW OF CASES, AND OF SUBCASES THE LAW OF CASES, AND OF SUBCASES, ctd P [ A | C ] = P [ A C ] P [ C ] = i P [ A | C i ] P [ C i | C ] . In words: The conditional probability for A to occur in case C is a weighted average of the conditional probabilities for A to occur in the various subcases; the weights are the conditional probabilities of the subcases given C . This rule is called the law of subcases . When C = S (and thus P [ C i | C ] = ), one gets P [ A ] = X i P [ A | C i ] P [ C i ] . 1 which is called the law of cases (or law of total probability). This method of computing conditional and unconditional probabilities is called conditioning on the cases , or just conditioning . 1 Eqn (3.4) on p.73 in Ross 3 / 13 Xinghua Zheng Lect 5: Law of Cases/Subcases

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THE LAW OF CASES, AND OF SUBCASES SEX BIAS IN GRADUATE ADMISSIONS 1973, University of California, Berkeley 44% of male applicants and 35% of female applicants were admitted — Gender bias? Comparison is made over the whole university – which departments were responsible for this bias? 4 / 13 Xinghua Zheng Lect 5: Law of Cases/Subcases
THE LAW OF CASES, AND OF SUBCASES SEX BIAS IN GRADUATE ADMISSIONS, ctd Male Female Number of Percent Number of Percent Major applicants admitted applicants admitted A 825 62% 108 83% B 560 63% 25 68% C 325 37% 593 34% D 417 33% 375 35% E 191 28% 393

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## This note was uploaded on 11/27/2011 for the course ISOM 3540 taught by Professor Zheu during the Spring '11 term at HKUST.

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lect5 - THE LAW OF CASES AND OF SUBCASES EXCHANGEABILITY...

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