Let X be a random variable following binomial distribution with parameters n = 6 and p = k. Further, 9P(X = 4) = P(X = 2). What is the value of k?

P(X = 4) = C(6,4)·k⁴·(1−k)² = 15k⁴(1−k)². P(X = 2) = C(6,2)·k²·(1−k)⁴ = 15k²(1−k)⁴. The condition 9P(X = 4) = P(X = 2) gives 9·15k⁴(1−k)² = 15k²(1−k)⁴, so 9k² = (1−k)². Taking square root: 3k = 1 − k (taking positive root since 0 < k < 1), so 4k = 1, k = 1/4.