55 sometimes we are given conditional probabilities

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55
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Sometimes we are given conditional probabilities and we need to compute the probability that several events occur simultaneously. Using the definition of conditional probability we can always write P A B P A | B P B so that, given P B and the conditional probability P A | B , we can find P A B . 56
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We can extend this to many events by applying this representation repeatedly: P A k A k 1 A 1 P A k | A k 1 A 1 P A k 1 A 1 P A k | A k 1 A 1 P A k 1 | A k 2 A 1 P A k 2 A 1 and so on, to get P A k A k 1 A 1 P A k | A k 1 A 1 P A k 1 | A k 2 A 1 P A 2 | A 1 P A 1 57
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By simply reversing the order of events we can also write the relationship – which is useful for sequential experiments – as P A 1 A 2 A k P A 1 P A 2 | A 1  P A k 1 | A k 2 A 1 P A k | A k 1 A 1 58
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