NDA Mathematics Question (2025) — Solution

Step-by-step solution

For a binomial distribution, the probability of r successes is given by P(X = r) = ^ n C_ r p^r q^ n-r , where q = 1 - p. Given n = 6 and p = k, we have q = 1 - k. The given condition is 9P(X = 4) = P(X = 2). Substituting the probabilities, we get: 9 ^ 6 C_ 4 k^4 (1-k)^2 = ^ 6 C_ 2 k^2 (1-k)^4 Since ^ 6 C_ 4 = ^ 6 C_ 2 = 15, we can cancel 15 k^2 (1-k)^2 from both sides (as k 0 and k 1): 9k^2 = (1-k)^2 Taking the square root on both sides (since k > 0 and 1-k > 0): 3k = 1 - k 4k = 1 k = 1 4 Now, we need to find P(X = 3): P(X = 3) = ^ 6 C_ 3 k^3 (1-k)^3 P(X = 3) = 20 ( 1 4 )^3 ( 3 4 )^3 P(X = 3) = 20 1 64 27 64 P(X = 3) = 540 4096 = 135 1024 Answer: 135 1024