Let X B(n, p) . If E(X) = 2 and Var (X) = 1 , then the probability of getting at most one success is .....

Step-by-step solution

For a binomial distribution X B(n, p) , the mean and variance are given by: E(X) = np = 2 Var (X) = npq = 1 Dividing the variance by the mean: q = npq np = 1 2 Since p + q = 1 , we get p = 1 2 . Substituting p in np = 2 : n ( 1 2 ) = 2 n = 4 The probability of getting at most one success is: P(X 1) = P(X = 0) + P(X = 1) P(X 1) = ⁴C₀ ( 1 2 )^0 ( 1 2 )^4 + ⁴C₁ ( 1 2 )^1 ( 1 2 )^3 P(X 1) = 1 16 + 4 16 = 5 16 P(X 1) = 0.3125 Answer: 0.3125