\begin{definition}[Definition:Binomial Distribution]
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ has the '''binomial distribution with parameters $n$ and $p$''' {{iff}}:
:$\Img X = \set {0, 1, \ldots, n}$
:$\map \Pr {X = k} = \dbinom n k p^k \paren {1 - p}^{n - k}$
where $0 \le p \le 1$.
Note that this distribution gives rise to a probability mass function satisfying $\map \Pr \Omega = 1$, because:
:$\ds \sum_{k \mathop \in \Z} \dbinom n k p^k \paren {1 - p}^{n - k} = \paren {p + \paren {1 - p} }^n = 1$
This is apparent from the Binomial Theorem.
It is written:
:$X \sim \Binomial n p$
\end{definition}