s4 - Math 4653: Elementary Probability: Spring 2007...

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Math 4653: Elementary Probability: Spring 2007 Homework #4. Problems and Solutions 1. Sec. 3.1: #8a): A hand of five cards contains two aces and three kings. The five cards are shuffled and dealt one by one, until an ace appears. Display in a table the distribution of the number of cards dealt. Solution. Let X be the number of cards dealt until an ace appears. We compute that P ( X = 1) = 2 5 , P ( X = 2) = 3 5 · 2 4 , P ( X = 3) = 3 5 · 2 4 · 2 3 , P ( X = 4) = 3 5 · 2 4 · 1 2 · 2 2 . These computations are summarized in the following table x 1 2 3 4 P ( X = x ) 2 / 5 3 / 10 1 / 5 1 / 10 2. Sec. 3.2: #14: A building has 10 floors above the basement. If 12 people get into an elevator at the basement, and each chooses a floor at random to get out, independently of the others, at how many floors do you expect the elevator to make a stop to let out one or more of these 12 people? Solution. Let X j = ± 1 if the elevator stops at floor j, 0 if the elevator does not stop at floor j. Then 10 j =1 X j is the number of floors at which the elevator makes a stop. Since the probability that the elevator fails to stop at a particular floor is (0 . 9) 12 , the probability that the elevators stops at a particular floor is 1 - (0 . 9) 12 , and E ( X j ) = 1 - (0 . 9) 12 . Therefore, E ( X ) = E ² 10 X j =1 X j ³ = 10 X j =1 E ( X j ) = 10 · [1 - (0 . 9) 12 ] 7 . 1757 . 3. Sec. 3.3: #2: Let Y be the number of heads obtained if a fair coin is tossed three times. Find the mean and variance of
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s4 - Math 4653: Elementary Probability: Spring 2007...

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