Revision: Binomial Distribution Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board

In a random experiment, if there are only two outcomes, success and failure, and the sum of the probabilities of these two outcomes is one, then any trial of such an experiment is known as a Bernoulli trial.

The probability distribution of the number of successes in an experiment consisting of n-Bernoulli trials obtained by the binomial expansion of (q + p )ⁿ is called the binomial distribution.

where p = probability of success and
q = probability of failure

\[P\left(X=r\right)=^{n}C_{r}p^{r}q^{n-r}\] is called probability function.

Trials of a random experiment are called Bernoulli trials if they satisfy the following

1. Each trial has exactly two outcomes: success or failure.

2. The probability of success remains the same in each trial.

The probability distribution of a random variable X, which represents the number of successes in n independent Bernoulli trials, each having probability of success p, is called the Binomial Distribution.

If q = 1 - p, then the probability function is given by 

\[P(X=x)=\binom{n}{x}p^xq^{n-x},\quad x=0,1,2,\ldots,n.\]

The probability of x successes, P(X = x), also denoted by P(x), is given by

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

This P(x) is called the probability function of the binomial distribution.

\[\mathrm{Mean~(\mu)=E(X)=np}\]

\[Variance(\sigma^2)=npq\]

\[\text{Standard deviation }(\sigma)=\sqrt{\mathrm{npq}}\]

Var (X ) = npq

E(X) = np

\[\sigma=\sqrt{npq}\]