Here, you will learn formula for binomial probability distribution in probability with example.

Let’s begin –

Suppose that we have an experiment such as tossing a coin or die repeatedly or choosing a marble from an urn repeatedly. Each toss or selection is called a trial. In any single trial there will be a probability associated with a particular event such as head on the coin, 4 on the die, or selection of a red marble. In some cases this probability will not change from one trial to the next(as in tossing a coin or die). Such trials are then said to be independent and are often called Bernoulli trials after James Bernoulli who investigated them at the end of the seventeenth century.

## Formula for Binomial Probability Distribution

Let p be the probability that an event will happen in any single Bernoulli trial(called the probability of success).Then q = 1 – p is the probability that the event will fail to happen in any single trial (called the probability of Failure). The probability that the event will happen exactly x times in n trials (i.e., x success and n – x failures will occur) is given by the probability function.

f(x) = P(X = x) = \(\binom{n}{x} p^x q^{n-x}\) = \(n!\over {x!(n – x)!}\) \(p^xq^{n-x}\)

where the random variable X denotes the number of success in n trials and x = 0, 1,…….,n.

Example : What is the probability of getting exactly 2 heads in 6 tosses of a fair coin?

Solution : The probability of getting exactly 2 heads in 6 tosses of a fair coin is

P(X = 2) = \(\binom{6}{2} ({1\over 2})^2 ({1\over 2})^{6-2}\)

= \(6!\over {2!4!}\) \(({1\over 2})^2 ({1\over 2})^{6-2}\)

= \({15}\over{64}\)

The discrete probability function is often called the binomial distribution since for x = 0, 1, 2,……,n, it corresponds to successive terms in the binomial expansion.

\((q + p)^n\) = \(q^n\) + \(\binom{n}{1} p q^{n-1}\) + \(\binom{n}{2} p^2 q^{n-2}\) + ……….+ \(p^n\) = \({\sum_{n=1}^{\infty}}\)\(\binom{n}{x} p^x q^{n-x}\)