Math135_hw6-soln

Math135_hw6-soln - Math 135, Fall 2011: HW 6 This homework...

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Math 135, Fall 2011: HW 6 This homework is due at the beginning of class on Wednesday October 19th, 2011 . You are free to talk with each other and get help. However, you should write up your own solutions and understand everything that you write. Problem 1 (p.217 #1) . A coin which lands heads with probability p is tossed repeatedly. Assuming independence of the tosses, find formulae for a) the probability that exactly 5 heads appear in the first 9 tosses; b) the probability that the first head appears on the 7th toss; c) the probability that the fifth head appears on the 12th toss; d) the probability that the same number of heads appear in the first 8 tosses as in the next 5 tosses. Part (a) is a binomial (n,p) probability, where n = 9. Let q = 1 - p . We get P (exactly 5 heads in first 9 tosses) = ± 9 5 ² p 5 q 4 . Part (b) is a geometric distribution; in order for the first head to appear at the 7th toss, the previous 6 tosses must be tails and the 7th toss must be heads. By independence, this is q 6 p . Part (c) is a negative binomial distribution. In order for the fifth head to appear on the 12th toss, the 12th toss must be a heads, and there must be exactly 4 heads in the previous 11 tosses. Thus P (5th head appears on 12th toss) = ± 11 4 ² p 4 q 7 p = ± 11 4 ² p 5 q 7 . For part (d), observe that by independence of trials, the random variables X 1 = [number of heads in the first 8 tosses] X 2 = [number of heads in the next 5 tosses] are independent. Also X 1 B (8 ,p ), and X 2 B (5 ,p ). Hence P (same # of H in first 8 as in next 5 tosses) = 5 X k =1 P ( X 1 = k,X 2 = k ) = 5 X k =1 P ( X 1 = k ) P ( X 2 = k ) = 5 X k =1 ± 8 k ² p k q 8 - k ± 5 k ² p k q 5 - k Problem 2 (p.218 #4) . A game is played in which three people each toss a fair coin to see if one of their coins shows a different face from the other two. The game ends if one of the three people has a coin which lands differently from the other two. 1
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a) What is the probability of the game ending after one round of play? b) What is the probability that the game lasts r rounds? c) What is the expected duration of play? SOLUTION. The probability of one of the coins landing differently from the other two is the com- plement of the probability that all three land heads or all three land tails. So P (game ends after one round) = 1 - ± 1 8 + 1 8 ² = 3 4 . The probability the game ends after one round is 3 4 . If game ends after r rounds, the first r - 1 rounds must have ended with no coin landing differ- ently from the others, and one coin landing differently from the other two on the rth round. By independence, this is P (game lasts r rounds) = ± 1 4 ² r - 1 ± 3 4 ² . Since the number of rounds of play is a geometric random variable with success probability
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Math135_hw6-soln - Math 135, Fall 2011: HW 6 This homework...

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