Which distribution models the number of successes in a fixed number of Bernoulli trials?

• A. Geometric distribution
• B. Binomial distribution X ~ Bin(n,p); mean np; variance np(1-p)
• C. Normal distribution Z ~ N(0,1)
• D. Poisson distribution with parameter λ

Answer: (B)

Counting how many successes you get when you perform a fixed number of independent Bernoulli trials with the same probability of success is modeled by the binomial distribution. If you perform n trials and each trial has a success probability p, the number of successes X follows X ~ Bin(n,p). Its mean is np and its variance is np(1-p), which exactly matches the description given in the question. This makes the binomial distribution the natural model for this scenario.

Geometric would track the number of trials needed to get the first success, which isn’t about a fixed number of trials. Normal is a continuous model and serves as an approximation for large n, not the exact discrete count of successes in a set number of trials. Poisson models the count of events in a fixed interval with a rate λ and can approximate binomial when n is large and p is small (with λ = np), but it’s not the exact description for a fixed n Bernoulli trial setup.