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.

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

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

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

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

To find _{n-1}C_{r-1} Calculate ((n-1)-(r-1))!
(n-1)-(r-1) = 8-3 = 5
((n-1)-(r-1))! = 5! = 120

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

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

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

To Solve _{n-1}C_{r-1} formula is used.
= 336/6 = 56

Find p^{r}.
= 0.5^{4} = 0.0625

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

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

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.

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