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# lec5 - Conditional Probability Example 1 A box has 5...

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Conditional Probability Example 1: A box has 5 computer chips. Two are defective. Two chips are selected from the box, one at a time. 1. Compute the probability that the second chip is defective. Again common sense tells us that P ( a chip is defective ) = 2 5 . Using probability theory, we have | | = # of ways to draw 2 chips (ordered) = (5)(4)=20 Define event A = the second chip is defective | A | = # of outcomes with second chip defective= (4)(2)=8 P ( second chip is defective ) = | A | | | = 8 20 = . 4 . 2. If we know that the first chip drawn is good, what is the probability that the second chip is defective? 1

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Conditional Probability (continued) When we obtain additional information about a probability experiment, we want to reassess the probabilities of events given the new information. Definition: The conditional probability of an event A given an event B is P ( A | B ) = P ( A B ) P ( B ) , provided P ( B ) ̸ = 0 . The above definition makes sense because the conditional probability of A given B is the fraction of outcomes in B that are also in A . An important implication of the definition is that P ( A B ) = P ( A | B ) P ( B ) = P ( B | A ) P ( A ) . Knowing two of the above probabilities allow us to compute the third. 2
Conditional Probability (continued) Compute the probability that the second chip is defective given that the first chip is good using the above definition. Let A =the first chip is good; B = the second chip is defective. Now we need to compute P ( B | A ) First, we have that P ( A ) = 3 5 The event A B is first chip is good and the second chip is defective.

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lec5 - Conditional Probability Example 1 A box has 5...

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