Binomial Distribution: Conditions and Probability Mass Function

The binomial distribution is the theoretical distribution that applies whenever an experiment consists of a fixed number of Bernoulli trials — trials that satisfy all four of these conditions:

1. Each trial has exactly two possible outcomes, conventionally called success and failure (pass/fail, defective/non-defective, heads/tails).
2. The number of trials, $n$, is fixed in advance.
3. The trials are independent — the outcome of one trial does not affect another.
4. The probability of success, $p$, is constant across all trials (and $q = 1 - p$ is the constant probability of failure).

Under these four conditions, if $X$ is the number of successes in $n$ trials, $X$ follows a binomial distribution, and the probability of getting exactly $x$ successes is given by its probability mass function:

where $\binom{n}{x} = \dfrac{n!}{x!\,(n-x)!}$ counts the number of different orders in which those $x$ successes can occur among the $n$ trials. Every factor in this formula plays a distinct role: $p^x$ is the probability of getting successes on exactly those $x$ trials, $q^{n-x}$ is the probability of failure on the remaining trials, and $\binom{n}{x}$ accounts for the fact that the $x$ successes could fall on any of the $\binom{n}{x}$ possible subsets of the $n$ trials.

A quick sanity check worth applying to every binomial problem: the probabilities for $x = 0, 1, \ldots, n$ must always add up to exactly $1$, since $\sum_{x=0}^{n} \binom{n}{x} p^x q^{n-x} = (p+q)^n = 1^n = 1$. This is a fast way to catch an arithmetic slip before submitting an answer. …