Binomial Distribution

The binomial distribution applies to events that can be described as a "success" if one outcome occurs or a "failure" if any other outcome occurs. There can be more than 2 outcomes, but it needs to be black and white in terms of success or failure.

The binomial distribution is given by the equation

\begin{align} P(X=r)= \left ( \begin{matrix}n\\ r\end{matrix}\right ) (1-p)^{n-r}p^r \end{align}


  • p is the probability that an event is successful (i.e. the probability of getting heads)
  • 1-p represent the probability that the event is a failure
  • r is the number of successful outcomes
  • n is the number of trials or attempts.

For example if we want to know the probability of getting 3 heads on a coin flip when flipping the coin a total of 5 times then:

\begin{align} P(X=3)= \left ( \begin{matrix}5\\ 3\end{matrix}\right ) (1-0.5)^{5-3}(0.5)^3 \end{align}

The coefficient $\left ( \begin{matrix}5\\ 3\end{matrix}\right )$ can be calculated with your calculator, see the page on Binomial Theorem.

Connection to Binomial Theorem

Yes, there is a remarkable similarity between a binomial distribution and a binomial expansion… Click below for more details.

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