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.

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-1C_r-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-1C_r-1 formula is used. = 336/6 = 56

Step 8:

Find p^r. = 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.