A binomial distribution arises from a sequence of independent Bernoulli trials.

Definition

If the following conditions hold:

  1. There is a sequence of \(n\) Bernoulli trials, each with two outcomes "success" and "failure", where \(n\) is a fixed constant,
  2. The results of all Bernoulli trials are independent of each other,
  3. The probability of "success" is the same for all trials, \(P(success) = \pi\),

then the total number of successes, \(X\), has a binomial distribution with parameters n and \(\pi\).

\[ X \;\; \sim \;\; \BinomDistn(n, \pi) \]

In most practical applications, the parameter \(\pi\) is an unknown constant, but occasionally we know its value.

Before using a binomial distribution, we must be able to argue from the context that the above three assumptions hold.