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Unformatted text preview: to form a partition of Ω if the events are mutually exclusive and Ω = E1 ∪ · · · ∪ Ek . Of course for a partition, P (E1 ) + · · · + P (Ek ) = 1. More generally, for any 48 CHAPTER 2. DISCRETE-TYPE RANDOM VARIABLES event A, the law of total probability holds because A is the union of the mutually exclusive sets AE1 , AE2 , . . . , AEk : P (A) = P (AE1 ) + · · · + P (AEk ). If P (Ei ) = 0 for each i, this can be written as P (A) = P (A|E1 )P (E1 ) + · · · + P (A|Ek )P (Ek ). Figure 2.8 illustrates the conditions of the law of total probability. E1 E2 A E3 E 4 ! Figure 2.8: Partitioning a set A using a partition of Ω. The definition of conditional probability and the law of total probability leads to Bayes’ formula for P (Ei |A) (if P (A) = 0) in simple form: P (AEi ) P (A|Ei )P (Ei ) = , P ( A) P (A) (2.13) P (A|Ei )P (Ei ) . P (A|E1 )P (E1 ) + · · · + P (A|Ek )P (Ek ) (2.14) P (Ei |A) = or in expanded form: P (Ei |A) = An important point about (2.13) is that it is a formula for P (Ei |A), whereas the l...
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