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Unformatted text preview: Math 408, Spring 2007 Midterm Exam 1 Solutions 1. Suppose A , B , and C are mutually independent events with probabilities P ( A ) = . 5, P ( B ) = 0 . 8, and P ( C ) = 0 . 3. Find the probability that exactly two of the events A,B,C occur. Solution. Let D denote the event exactly two of A,B,C . Note that D is a union of the three sets A B C , A B C , A B C , and that these three sets are mutually disjoint. Thus, P ( D ) = P ( A B C ) + P ( A B C ) + P ( A B C ) By the independence of A,B,C , the probabilities of the intersections on the right are equal to the products of the individual probabilities, so P ( D ) = P ( A ) P ( B ) P ( C ) + P ( A ) P ( B ) P ( C ) + P ( A ) P ( B ) P ( C ) = 0 . 5 . 8 (1- . 3) + 0 . 5 (1- . 8) . 3 + (1- . 5) . 8 . 3 = . 43 Alternative solution: Draw a Venn diagram of the three sets A , B , C . The event D consists of the pairwise overlaps A B , A C , and B C , but excluding the triple overlap, i.e., A B C . If the probabilities of the pairwise overlaps are added, the triple overlap is counted three times, so it has to be subtracted again three times. Thus, P ( D ) = P ( A B ) + P ( A C ) + P ( B C )- 3 P ( A B C ) = 0 . 5 . 8 + 0 . 5 . 3 + 0 . 8 . 3- 3 . 5 . 8 . 3 = . 43 2. Suppose that 25% of all calculus students get an A, and that students who had an A in calculus are 50% more likely to get an A in Math 408 as those who had a lower grade in calculus. If a student who received an A in Math 408 is chosen at random, what is the probability that he/she also received an A in calculus?...
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