The binomial distribution

Learning objectives

Justify whether a random variable is binomial by checking for a fixed number of independent trials, two outcomes on each trial and the same probability of success on every trial

Calculate the probability of exactly k successes with P(X = k) = ⁿCₖ pᵏ(1 − p)ⁿ⁻ᵏ, and the probability of at most or at least k successes, from the formula or with technology

Calculate the mean np and the standard deviation √(np(1 − p)) of a binomial random variable, and interpret both in context

Estimate a binomial probability from the results of a simulation, and compare the estimate with the value given by the formula

Explain why sampling without replacement breaks the independence of the trials, and use the 10% condition to justify a binomial model when the sample is a small part of the population

Interpret a binomial probability in context, and use it to judge whether an observed count would be surprising if the stated probability of success were correct