hw1_sol - EE 351K Probability, Statistics, and Random...

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EE 351K Probability, Statistics, and Random Processes SPRING 2008 Instructor: Shakkottai/Vishwanath { shakkott,sriram } @ece.utexas.edu Homework 1 Problem 1 We are given that P ( A ) = 0 . 55 , P ( B c ) = 0 . 45 , and P ( A B ) = 0 . 25 . Determine P ( B ) and P ( A B ) . Solution : We have P ( B ) = 1 - P ( B c ) = 1 - 0 . 45 = 0 . 55 . Also, by rearranging the formula P ( A B ) = P ( A ) + P ( B ) - P ( A B ) , we obtain P ( A B ) = P ( A ) + P ( B ) - P ( A B ) = 0 . 55 + 0 . 55 - 0 . 25 = 0 . 85 . Problem 2 Let A and B be two sets. (a) Show that ( A c B c ) c = A B and ( A c B c ) c = A B . (b) Consider rolling a six-sided die once. Let A be the set of outcomes where an odd number comes up. Let B be the set of outcomes where a 1 or a 2 comes up. Calculate the sets on both sides of the equalities in part (a), and verify that the equalities hold. Solution : (a) See scanned attachment. (b) We have A = { 2 , 3 , 5 } , B = { 3 , 6 } . Thus, A c B c = { 2 , 4 , 6 } ∩ { 3 , 4 , 5 , 6 } = { 4 , 6 } , ( A c B c ) c = { 1 , 2 , 3 , 5 } , A B = { 1 , 2 , 3 , 5 } , so the first equality is verified. Similarly, A
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hw1_sol - EE 351K Probability, Statistics, and Random...

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