# Binomial Distribution Tutorial

## Binomial Distribution Tutorial

##### Definition:

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.

#### Formula:

P(X = r) = nCr p r (1-p)n-r
###### where,

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

##### Example:

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

###### Step 1:

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

###### Step 2:

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

###### Step 3:

Find pr. pr = 0.57 = 0.0078125

###### Step 4:

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

###### Step 5:

Find (1-p)n-r. = 0.55 = 0.03125

###### Step 6:

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