Bernoulli trial is also known as binomial trial where only two outcomes of a given experiment is possible. If a flip a coin, only two outcomes are possible, that is, head and tail. Hence, flipping of coin is a Bernoulli trial. If we roll a dice six outcomes are possible, that is, $1, 2, 3, 4, 5, 6$ and hence, rolling of a dice is not a Bernoulli trial. To get the probability of the outcomes of a Bernoulli trial, binomial probability formula is used. Some real life examples of a Bernoulli trial are if a bulb is on or off, if a question is answered correctly or not, if a student has passed or failed.

## Definition

A trial where only two outcomes are possible is known as Bernoulli trial. Trials like flipping of coin can be termed as Bernoulli trial.

Suppose, we are flipping a coin $4$ times and we want to know the probability of getting a head $3$ out of $4$ times.

The probability of getting a head, $p$ = $0.5$

The probability of getting a tail, $q$ = $0.5$

The probability of getting a head $3$ out of $4$ times will be = $pppq$ = $0.5\ \times\ 0.5\ \times\ 0.5\ \times\ 0.5$

But
out of $4$ times, which trials will give head. For that, we will use
combination. The $3$ times when head will come out of $4$ times can be
arranged in $C_{3}^{4}$ ways.

Hence, probability of getting $3$ heads out of $4$ will be $C_{3}^{4}(0.5)^3(0.5)^1$

## Formula

Suppose, n Bernoulli trials are made then the probability of getting r successes in n trials can be given by the formula,

$P(r) = C_{r}^{n}p^rq^{n-r}$

The term $\frac{n!}{r!(n - r)!}$ is known as binomial coefficient.For example, if a student is attempting five true or false questions, then find the probability of getting $3$ correct answers.

Probability of success, $p$ = $0.5$

Probability of failure, $q$ = $0.5$

Probability of getting $3$ correct answers = $C_{3}^{5}(0.5)^3(0.5)^2$ = $0.3125$.

## Bernoulli Trial Conditions

The characteristics of a Bernoulli trial are:

**1)** It can have only two outcomes, that can be labelled as success and failure.

**2)** Probability of success and failure remains same through each trial.

**3)** The trials are independent of each other.

**4)** Number of trials are fixed.

If $p$ is the probability of success, then the probability of failure, $q = 1 - p$.

## Bernoulli Trial Binomial Distribution

If a trial is done n times, then to find the probability of success happening r times, probability distribution formula is used. The Bernoulli trial binomial distribution formula is given as:

$P(X = r) = C_{r}^{n}p^rq^{n-r}$This formula can give solutions to problems like the probability of r success in n trials, probability of at least one success in n trials probability of no success in n trials, probability of at least one failure in n trials and probability of at most r success in n trials.