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131A_1_ds1_sol_2010fall

131A_1_ds1_sol_2010fall - EE 131A Probability Instructor...

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EE 131A Discussion Set 1 Probability Wednesday, September 29, 2010 Instructor: Lara Dolecek and Friday, October 1, 2010 Reading: Chapter 2 of Probability, Statistics, and Random Processes by A. Leon-Garcia 1. Venn diagram. Problem 2.13, page 82 of ALG Solution : Using the Venn diagram in Figure 1, the event “exactly one of the events A and B occurs” can be expressed as ( A B c ) ( A c B ) Figure 1: Decomposition of A B into three disjoint sets. 2. Balls and buckets. Problem 2.50, page 86 of ALG Solution : The possible ways to cover all the buckets =5! The possible placements of 5 balls in 5 buckets=5 5 Therefore,the probability that each bucket has a ball= 5! 5 5 3. Bonferroni’sinequality. Suppose A and B are two events. Use the axioms of probability to prove the following: P ( A B ) P ( A ) + P ( B ) - 1 Proof : Using Corollary 5, P ( A B ) = P ( A ) + P ( B ) - P ( A B ) Using Corollary 2, P ( A B ) 1 Therefore, P ( A ) + P ( B ) - P ( A B ) 1 = P ( A B ) P ( A ) + P ( B ) - 1 Show that the probability that one and only one of the events A or
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