3Conditional_Probability (1)

3Conditional_Probability (1) - Chapter 3 Conditional...

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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 is equal to Pr( | B ) = Pr( 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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3Conditional_Probability (1) - Chapter 3 Conditional...

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