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First Course in Probability, A (7th Edition)

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Fourth Homework Set — Solutions Chapter 3 Problem 57 (a) P (original price after two days) = ( 2 1 ) p (1 - p ) = 2 p (1 - p ) (b) P (increase by one after three days) = ( 3 2 ) p 2 (1 - p ) = 3 p 2 (1 - p ) (c) P (increase on first day | increase by one after three days) = p · 2 p (1 - p ) 3 p 2 (1 - p ) = 2 3 Problem 59 (a) P ( HHHH ) = p 4 (b) P ( THHH ) = p 3 (1 - p ) (c) The pattern HHHH can only occur before THHH if the first four coin flips come up heads. Hence, P ( THHH occurs before HHHH ) = 1 - p 4 . Problem 64 Let E be the event that the wife answers correctly, and let F be the event that the husband answers correctly. (a) If only one of them answers, then the probability of a correct answer is P ( E ) = P ( F ) = p . (b) P (correct answer) = P ( EF ) + 1 2 · 2 · p (1 - p ) = p 2 + p - p 2 = p Problem 66 Let E i be the event that the i -th switch is on. (a) P (current flows from A to B ) = ( P ( E 1 E 2 ) + P ( E 3 E 4 ) - P ( E 1 E 2 E 3 E 4 )) P ( E 5 ) = ( p 1 p 2 + p 3 p 4 - p 1 p 2 p 3 p 4 ) p 5 1
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(b) P (current flows from A to B ) = P ( E 1 E 4 E 1 E 3 E 5 E 2 E 5 E 2 E 3 E 4 ) = p 1 p 4 + p 1 p 3 p 5 + p 2 p 5 + p 2 p 3 p 4 - p 1 p 3 p 4 p 5 - p 1 p 2 p 4 p
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