Statistical Sampling

Binomial distribution

Assume that we have an experiment which can be performed with a success rate $p$ , when all experiments are performed independently of each other. Assume the experiements is performed $n$ times. Define then the stochastic variable $K$ which represents how many of the $n$ experiments are successful. This variable then follows a binomial distribution.

The probability that $k$ experiments are successful, independently is their order (!), equals $\binom{n}{k} p^k (1 - p)^{n - k}$ , which accounts for $k$ successes, $n - k$ failures, and where $\binom{n}{k}$ are binomial coefficients, which count the number of distinct ways in which the successes and failures can be ordered.

As an example, for two experiements ( $n = 2$ ), there is one way to order two successes and two failures, thus $\binom{2}{0} = \binom{2}{2} = 1$ while for one success and one failure, there are two ways two order them, thus $\binom{2}{1} = 2$ . As such, one has $P(K = 0) = (1 - p)^2$ and $P(K = 2) = p^2$ , while $P(K = 1) = 2p(1 - p)$ .