A statistical experiment having successive independent trials having two possible outcomes (such as success and failure; true and false; yes and no; right and wrong etc.) and probability of success is equal for each trial, while this kind of experiment is repeated a fixed number of times (say $n$ times) is called *Binomial Experiment*, Each *trial* of this *Binomial experiment* is known as *Bernoulli trial* (a trial which is a single performance of an experiment), for example. There are four properties of *Binomial Experiment*.

- Each
*trial* of *Binomial Experiment* can be classified as *success* or *failure*.
- The
*probability of success* for each trial of the experiment is equal.
- Successive
*trials are independent*, that is, the occurrence of one outcome in an experiment does not affect occurrence of the other.
- The experiment is
*repeated* *a* *fixed number of times*.

Let a *discrete random variable*, which denotes the *number of successes* of a *Binomial Experiment* (we call this binomial random variable). The *random variable* assume isolated values as $X=0,1,2,\cdots,n$. The *probability distribution* of *binomial random variable* is termed as *binomial probability distribution*. It is a *discrete probability distribution*.

**Binomial Probability Mass Function**

The probability function of binomial distribution is also called binomial probability mass function and can be denoted by $b(x, n, p)$, that is, a binomial distribution of random variable $X$ with $n$ (given number of trials) and $p$ (probability of success) as parameters. If $p$ is the probability of success (alternatively $q=1-p$ is probability of failure such that $p+q=1$) then probability of exactly $x$ success can be found from the following formula,

\begin{align}

b(x, n, p) &= P(X=x)\\

&=\binom{n}{x} p^x q^{n-x}, \quad x=0,1,2, \cdots, n

\end{align}

where $p$ is probability of success of a single trial, $q$ is probability of failure and $n$ is number of independent trials.

The formula gives probability for each possible combination of $n$ and $p$ of binomial random variable $X$. Note that it does not give $P(X <0)$ and $P(X>n)$. Binomial distribution is suitable when $n$ is small and is applied when sampling done is with replacement.

\[b(x, n, p) = \binom{n}{x} p^x q^{n-x}, \quad x=0,1,2,\cdots,n,\]

is called Binomial distribution because its successive terms are exactly same as that of binomial expansion of

\begin{align}

(q+p)^n=\binom{0}{0} p^0 q^{n-0}+\binom{n}{1} p^1 q^{n-1}+\cdots+\binom{n}{n-1} p^n q^{n-(n-1)}+\binom{n}{n} p^n q^{n-n}

\end{align}

$\binom{n}{0}, \binom{n}{1}, \binom{n}{2},\cdots, \binom{n}{n-1}, \binom{n}{n}$ are called Binomial coefficients.

Note that it is necessary to describe the limit of the random variable otherwise it will be only the mathematical equation not the probability distribution.