# Binomial Probability

Binomial probability refers to the probability of exactly $x$ successes on $n$ repeated trials in an experiment which has two possible outcomes (commonly called a binomial experiment).

If the probability of success on an individual trial is $p$ , then the binomial probability is ${}_{n}{C}_{x}\cdot {p}^{x}\cdot {\left(1-p\right)}^{n-x}$ .

Here ${}_{n}{C}_{x}$ indicates the number of different combinations of $x$ objects selected from a set of $n$ objects. Some textbooks use the notation $\left(\begin{array}{c}n\\ x\end{array}\right)$ instead of ${}_{n}{C}_{x}$ .

Note that if $p$ is the probability of success of a single trial, then $\left(1-p\right)$ is the probability of failure of a single trial.

Example:

What is the probability of getting $6$ heads, when you toss a coin $10$ times?

In a coin-toss experiment, there are two outcomes: heads and tails. Assuming the coin is fair , the probability of getting a head is $\frac{1}{2}$ or $0.5$ .

The number of repeated trials: $n=10$

The number of success trials: $x=6$

The probability of success on individual trial: $p=0.5$

Use the formula for binomial probability.

${}_{10}{C}_{6}\cdot {\left(0.5\right)}^{6}\cdot {\left(1-0.5\right)}^{10-6}$

Simplify.

$\approx 0.205$

If the outcomes of the experiment are more than two, but can be broken into two probabilities $p$ and $q$ such that $p+q=1$ , the probability of an event can be expressed as binomial probability.

For example, if a six-sided die is rolled $10$ times, the binomial probability formula gives the probability of rolling a three on $4$ trials and others on the remaining trials.

The experiment has six outcomes. But the probability of rolling a $3$ on a single trial is $\frac{1}{6}$ and rolling other than $3$ is $\frac{5}{6}$ . Here, $\frac{1}{6}+\frac{5}{6}=1$ .

The binomial probability is:

${}_{10}{C}_{4}\cdot {\left(\frac{1}{6}\right)}^{4}\cdot {\left(1-\frac{1}{6}\right)}^{10-4}$

Simplify.

$\approx 0.054$