Binomial Distribution Questions with Solutions

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two outcomes (success or failure) and the probability of success p stays the same. Examples include the number of heads in ten coin tosses or the number of defective items in a sample. The probability of exactly r successes in n trials is P(X = r) = nCr p^r q^(n-r), where q = 1 - p. The mean is np and the variance is npq. Exam questions test finding the probability of exactly, at least or at most r successes, and finding n, p or q when the mean and variance are given. For "at least one" questions, use the complement: P(X >= 1) = 1 - q^n. Since the mean np is always greater than the variance npq, an option that gives a variance larger than the mean can be ruled out straight away. Practise the binomial distribution questions below, each with a step-by-step solution.