Theorem 16

Let p be the probability that event A will occur on a single trial. Then, in a series of n statistically independent observations or trials, the probability that event A occurs exactly k times is given by the binomial frequency function as

$$\text{Pr} (A \text{ occurs $k$\space times in $n$\space trials}) = \begin{pmatrix} n \\ k \end{pmatrix} p^k (1 - p)^{n-k}.$$

Proof: The proof is given in any elementary statistics text.