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By Pearn W. L., Lin G. H.

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How should we deal with “independent” repetitions of the same experiment, such as tossing a coin or rolling a die? More generally, how can {(Ω1 , F1 , P1 ), (Ω2 , F2 , P2 ), . } be probability spaces that correspond to independent experiments? Without constructing the whole of (Ω, F, P ) that corresponds to all these experiments in sequence, the key step is to have P (A1 ∩ A2 ∩ · · · ∩ An ) = P1 (A1 )P2 (A2 ) · · · Pn (An ), whenever each Ai belongs to Fi , i = 1, 2, 3, . . , and for all values of n.

99 %? Higher? ) that they use to interpret this time-honoured phrase. 1) Proof Because P (A∩Bj ) can be written both as P (Bj |A)P (A) and as P (A|Bj )P (Bj ), the first equality is immediate. 5 to give an alternative form of P (A). Take the special case when there are just two alternatives, B1 and B2 in this partition, so that B2 = B1c . The ratio P (B1 )/P (B2 ) is then termed the odds ratio of B1 , as it compares the chance B1 happens to the chance it does not happen. 2) In a Court, if B1 represents a Guilty verdict, this formula tells us how each piece of evidence A should influence our opinions.

Higher? ) that they use to interpret this time-honoured phrase. 1) Proof Because P (A∩Bj ) can be written both as P (Bj |A)P (A) and as P (A|Bj )P (Bj ), the first equality is immediate. 5 to give an alternative form of P (A). Take the special case when there are just two alternatives, B1 and B2 in this partition, so that B2 = B1c . The ratio P (B1 )/P (B2 ) is then termed the odds ratio of B1 , as it compares the chance B1 happens to the chance it does not happen. 2) In a Court, if B1 represents a Guilty verdict, this formula tells us how each piece of evidence A should influence our opinions.

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