Which distribution models the number of successes in a fixed number of trials with constant probability p?

• A. Normal distribution
• B. Poisson distribution
• C. Exponential distribution
• D. Binomial distribution

Answer: (D)

Counting how many successes occur in a fixed number of independent trials, each with the same probability of success, is described by the binomial distribution. If you perform n trials and each has probability p of success, the number of successes X follows Binomial(n, p). The probability of getting exactly k successes is P(X = k) = C(n, k) p^k (1 − p)^(n − k). The mean is np and the variance is np(1 − p). This model fits because you have a fixed number of trials and a constant chance of success on each trial, with independent outcomes. Other distributions describe different situations: the Poisson counts events in a fixed interval with a constant rate (often as a limit of the binomial when n is large and p is small), the exponential models waiting time between events in continuous time, and the normal distribution is a continuous approximation that can approximate the binomial under certain conditions.