# Proof: Probability mass function of the binomial distribution

**Index:**The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate discrete distributions ▷ Binomial distribution ▷ Probability mass function

**Theorem:** Let $X$ be a random variable following a binomial distribution:

Then, the probability mass function of $X$ is

\[\label{eq:bin-pmf} f_X(x) = {n \choose x} \, p^x \, (1-p)^{n-x} \; .\]**Proof:** A binomial variable is defined as the number of successes observed in $n$ independent trials, where each trial has two possible outcomes (success/failure) and the probability of success and failure are identical across trials ($p$, $q = 1-p$).

If one has obtained $x$ successes in $n$ trials, one has also obtained $(n-x)$ failures. The probability of a particular series of $x$ successes and $(n-x)$ failures, when order does matter, is

\[\label{eq:bin-prob} p^x \, (1-p)^{n-x} \; .\]When order does not matter, there is a number of series consisting of $x$ successes and $(n-x)$ failures. This number is equal to the number of possibilities in which $x$ objects can be choosen from $n$ objects which is given by the binomial coefficient:

\[\label{eq:bin-coeff} {n \choose x} \; .\]In order to obtain the probability of $x$ successes and $(n-x)$ failures, when order does not matter, the probability in \eqref{eq:bin-prob} has to be multiplied with the number of possibilities in \eqref{eq:bin-coeff} which gives

\[\label{eq:bin-pmf-qed} p(X=x|n,p) = {n \choose x} \, p^x \, (1-p)^{n-x}\]which is equivalent to the expression above.

**∎**

**Sources:**

**Metadata:**ID: P97 | shortcut: bin-pmf | author: JoramSoch | date: 2020-05-11, 22:35.