Isye 2027

9 box 1 has three black and two white box 1 box 2

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

Ask a homework question - tutors are online