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s3 - Math 4653 Elementary Probability Spring 2007...

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Math 4653: Elementary Probability: Spring 2007 Homework #3. Problems and Solutions 1. Sec. 2.4: #2: Find Poisson approximations to the probabilities of the following events in 500 independent trials with probability 0 . 02 of success on each trial: a) 1 success; b) 2 or fewer successes; c) more than 3 successes. Solution. Let X be the number of successes. The actual distribution of X is a binomial distribution with n = 500 and p = 0 . 02. Therefore, μ = np = 10, and the Poisson distribution provides a good approximation. a ) P ( X = 1) e - 10 10 1 1! 0 . 0004540 . b ) P ( X 2) 2 k =0 e - 10 10 k k ! = e - 10 (1 + 10 + 50) 0 . 0027694 . c ) P ( X > 3) 1 - 3 k =0 e - 10 10 k k ! = 1 - e - 10 1 + 10 + 50 + 500 3 0 . 98966 . 2. Sec. 2.4: #6a): A box contains 1000 balls, of which 2 are black and the rest are white. Which of the following is most likely to occur in 1000 draws with replacement from the box? fewer than 2 black balls, exactly 2 black balls, more than 2 black balls. Solution. Let X be the number of black balls drawn. The actual distribution of X is a binomial distribution with n = 1000 and p = 0 . 002. Therefore, μ = np = 2 , and the Poisson distribution provides a good approximation. We compute: P ( X < 2) 1 k =0 e - 2 2 k k ! = e - 2 (1 + 2) 0 . 4060 , P ( X = 2) e - 2 2 2 2! = e - 2 · 2 0 . 2707 , P ( X > 2) = 1 - P ( X < 2) - P ( X = 2) 1 - 0 . 4060 - 0 . 2707 = 0 . 3233 . Therefore, of the three events, the most likely is “fewer than 2 black balls” .

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s3 - Math 4653 Elementary Probability Spring 2007...

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