What is a binomial distribution?

• A. A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes.
• B. It describes a continuous variable distribution
• C. It describes the distribution of sample means
• D. It describes the distribution of variances across samples

Answer: (A)

Binomial distribution counts how many times a binary outcome occurs in a fixed number of independent trials, with the same chance of success on every trial. It is discrete, allowing values from 0 to n, where n is the number of trials and p is the probability of success on each trial. The probability of exactly k successes is P(X = k) = C(n, k) p^k (1 − p)^(n − k). This fits situations like flipping a coin n times and counting how many heads you get, or counting how many items pass quality control in a batch. The other descriptions refer to different kinds of distributions: a continuous variable distribution, the sampling distribution of the mean, or the sampling distribution of variances, none of which capture the count of successes in fixed independent trials.