The Binomial Distribution is one of the discrete probability distribution. It is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are appropriately labeled **Success** and **Failure**. The Binomial Distribution is used to obtain the probability of observing r successes in n trials, with the probability of success on a single trial denoted by p.

n = Number of events
r = Number of successful events.
p = Probability of success on a single trial.
_{n}C_{r} = ( n! / (n-r)! ) / r!
1-p = Probability of failure.

Toss a coin for 12 times. What is the probability of getting exactly 7 heads.

Here, Number of trials n = 12 Number of success r = 7 since we define getting a head as success Probability of success on any single trial p = 0.5

To Calculate _{n}C_{r} formula is used.
_{n}C_{r} = ( n! / (n-r)! ) / r!
= ( 12! / (12-7)! ) / 7!
= ( 12! / 5! ) / 7!
= ( 479001600 / 120 ) / 5040
= ( 3991680 / 5040 )
= 792

Find p^{r}.
p^{r} = 0.5^{7}
= 0.0078125

To Find (1-p)^{n-r} Calculate 1-p and n-r.
1-p = 1-0.5 = 0.5
n-r = 12-7 = 5

Find (1-p)^{n-r}.
= 0.5^{5} = 0.03125

Solve P(X = r) = _{n}C_{r} p^{ r }(1-p)^{n-r}
= 792 * 0.0078125 * 0.03125
= 0.193359375
The probability of getting exactly 7 heads is 0.19

This tutorial will help you to calculate the Binomial Distribution problems.