\begin{eqnarray*} {n \choose x} = \displaystyle \frac {n!}{x!(n-x)!} \\ \end{eqnarray*}

The number of successes in n independent Bernoulli trials has a binomial distribution.

n independent Bernoulli trials

• There are n independent trials
• Each trial can result in one of two possible outcomes, labelled success and failure.
  • success can be a bad thing – tire blow-up.
• P(success) = p,
• P(failure) = 1-p

\begin{eqnarray*} P(X=x) = _{n}C_{x} \cdot p^{x} \cdot (1-p)^{n-x}, \;\; \text{for} \;\; x = 0, 1, 2, . . ., n. \\ \end{eqnarray*}

\begin{eqnarray*} X \sim B(n, p) \\ \end{eqnarray*}