Suppose an event occurs with probability \(p\text{.}\) If we perform \(n\) independent trials, let \(S\) be the random variable which counts the number of trials in which the event occurred. Then \(S\) has the binomial distribution with parameters \(n\) and \(p\text{.}\) We’ll write \(S \sim \Bin(n, p)\) to denote this. For each value \(0 \leq k \leq n\text{,}\) we’ll write \(b(k)\) for \(\Pr(S = k)\text{.}\) (If we want to keep track of the parameter values, we may write \(b(k; n, p)\text{.}\))