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Unformatted text preview: Chapter 3 Conditional Probability Conditional probability provides a way to compute the likelihood of an event based on partial information . This is a powerful concept that is used exten sively throughout engineering with applications to decision making, networks, communications and many other fields. 3.1 Conditioning on Events We begin our treatment of conditional probability with an illustrative exam ple. The intuition gained through this simple exercise is then generalized by introducing a formal definition for this important concept. Example 6. The rolling of a fair die is an experiment with six equally likely outcomes. As such, the probability of obtaining any of the outcomes is 1 / 6 . However, if we are told that the upper face features an odd number, then only three possibilities remain, namely { 1 , 3 , 5 } . These three outcomes had equal probabilities before the additional informa tion was revealed. It then seems natural to assume that they remain equally likely afterwards. In particular, it is reasonable to assign a probability of 1 / 3 to each of the three outcomes that remain possible candidates after learning the new information. Note that we can express the probability of getting a three given that the outcome is an odd number as Pr(3 ∩ { 1 , 3 , 5 } ) Pr( { 1 , 3 , 5 } ) = Pr(3) Pr( { 1 , 3 , 5 } ) = 1 3 . 20 3.1. CONDITIONING ON EVENTS 21 Figure 3.1: Partial information about the outcome of an experiment may change the likelihood of events. The resulting values are known as conditional probabilities. Having reviewed a specific situation, we now turn to the more encompassing setting. Let B be an event such that Pr( B ) > 0. A conditional probability law assigns to every event A a number Pr( A  B ), termed the conditional probability of A given B , such that Pr( A  B ) = Pr( A ∩ B ) Pr( B ) . (3.1) We can show that the set of conditional probabilities { Pr( A  B ) } specifies a valid probability law, as defined in Section 2.2. For every event A , we have Pr( A  B ) = Pr( A ∩ B ) Pr( B ) ≥ . That is, Pr( A  B ) is nonnegative. The probability of the entire sample space Ω is equal to Pr(Ω  B ) = Pr(Ω ∩ B ) Pr( B ) = Pr( B ) Pr( B ) = 1 . If A 1 , A 2 , . . . is a sequence of disjoint events, then A 1 ∩ B, A 2 ∩ B, . . . is also a sequence of disjoint events and Pr ∞ k =1 A k B = Pr (( ∞ k =1 A k ) ∩ B ) Pr( B ) = Pr ( ∞ k =1 ( A k ∩ B )) Pr( B ) = ∞ k =1 Pr( A k ∩ B ) Pr( B ) = ∞ k =1 Pr( A k  B ) , 22 CHAPTER 3. CONDITIONAL PROBABILITY where the third equality follows from the third axiom of probability applied to the set ∞ k =1 ( A k ∩ B ). Thus, the conditional probability law defined by (3.1) satisfies the three axioms of probability....
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This note was uploaded on 03/30/2010 for the course ECEN 303 taught by Professor Chamberlain during the Fall '07 term at Texas A&M.
 Fall '07
 Chamberlain

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