\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*}

A balanced dice is rolled 3 times. What is probability a 5 comes up exactly twice?

p = 1/6
n = 3
x = 2

\begin{eqnarray*} P(X=2) & = & {{3} \choose {2}} \left(\frac{1}{6}\right)^{2} \left(\frac{5}{6}\right)^{3-2} \\ & = & 0.0694 \end{eqnarray*}

> dbinom(2, 3, 1/6)
[1] 0.06944444
> 

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