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.

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.

$\approx 0.054$