Let X be a random variable following binomial distribution with parameters n = 5 and p = k. Further, P(X = 1) = 0.4096 and P(X = 2) = 0.2048. What is the value of k?

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Correct answer: A. 0.2

Given X follows a binomial distribution with parameters n = 5 and p = k. The probability mass function is given by P(X = x) = ^ n C_ x p^ x (1-p)^ n-x . For x = 1: P(X = 1) = ^ 5 C_ 1 k^ 1 (1-k)^ 4 = 5k(1-k)^ 4 = 0.4096 For x = 2: P(X = 2) = ^ 5 C_ 2 k^ 2 (1-k)^ 3 = 10k^ 2 (1-k)^ 3 = 0.2048 Dividing P(X = 1) by P(X = 2), we get: P(X = 1) P(X = 2) = 5k(1-k)^ 4 10k^ 2 (1-k)^ 3 0.4096 0.2048 = 1-k 2k 2 = 1-k 2k 4k = 1 - k 5k = 1 k = 0.2

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