# Negative Binomial Distribution Tutorial

## Negative Binomial Distribution Tutorial

##### Definition:

The probability distribution of a Negative Binomial random variable is called a Negative Binomial Distribution. It is also known as the Pascal distribution or Polya distribution. Suppose we flip a coin repeatedly and count the number of heads (successes). If we continue flipping the coin until it has landed 2 times on heads, we are conducting a Negative Binomial Experiment.

#### Formula:

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

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

##### Example:

Find the probability that a man flipping a coin gets the fourth head on the ninth flip.

###### Step 1:

Here, Number of trials n = 9 (because we flip the coin nine times). Number of successes r = 4 (since we define Heads as a success). Probability of success for any coin flip p = 0.5

###### Step 2:

Find n-1 and r-1. n-1 = 9-1 = 8 r-1 = 4-1 = 3

###### Step 3:

To find n-1Cr-1 Calculate ((n-1)-(r-1))! (n-1)-(r-1) = 8-3 = 5 ((n-1)-(r-1))! = 5! = 120

###### Step 4:

Find (n-1)! = 8! = 40320

###### Step 5:

Find (r-1)! = 3! = 6

###### Step 6:

Find (n-1)! / ((n-1)-(r-1))! = 40320/120 = 336

###### Step 7:

To Solve n-1Cr-1 formula is used. = 336/6 = 56

###### Step 8:

Find pr. = 0.54 = 0.0625

###### Step 9:

To Find (1-p)n-r Calculate 1-p and n-r. 1-p = 1-0.5 = 0.5 n-r = 9-4 = 5

###### Step 10:

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

###### Step 11:

Calculate Negative Binomial Distribution. = 56x0.0625x0.03125 = 0.109375 The probability that the coin will land on heads for the fourth time on the ninth coin flip is 0.1094.

#### Related Calculator:

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