The Negative Binomial Distribution

A negative binomial distribution tells us how many trials (k) are required until we obtain the rth “success.”

Given that the probability of success in a given trial equals p, we can find the probability that we will obtain the rth success on the kth trial using the formula:

P(obtain rth success on kth trial) = k-1Cr-1 * pr * (1-p)k – r

The mean of the negative binomial distribution is μ = r/p

Let’s walk through some examples to get a better understanding of the negative binomial distribution.

Examples Using the Negative Binomial Distribution

Example 1: Mike makes 90% of his free-throw attempts. What is the probability that he makes his second free-throw on his fifth shot?

Step 1: Identify the probability of success on a given shot (p), the total number of shots (k), and the number of shots until the second make (r).

The probability that Mike makes a given free-throw is p = 0.9

We want to know if he will make his = 2nd shot on his = 5th attempt.

Step 2: Plug these numbers into the negative binomial formula or a negative binomial calculator.

Using the formula:

P(obtain rth success on kth trial) = k-1Cr-1 * pr * (1-p)k – r

P(obtain second make on fifth shot) = 5-1C2-1 * (.9)2 * (1-.9)5 – 2

P(obtain second make on fifth shot) = (4) * (.81) * (.001) = .00324

Using the calculator:

Plug the following numbers into the Negative Binomial Distribution Calculator:


The probability that Mike makes his second shot on his fifth attempt is .00324.

Example 2: At a certain university, 10% of all students are left-handed. If we randomly survey 100 students from this university, what is the probability that the 10th student we survey is the third left-handed student we encounter?

Step 1: Identify the probability that a given student is left-handed (p), the total number of students to survey until the third left-handed student (k), and the number left-handed students (r).

The probability that a given student is left-handed is p = 0.1

We want to know if the = 10th student is the = 3rd left-handed student we encounter.

Step 2: Plug these numbers into the negative binomial formula or a negative binomial calculator.

Using the formula:

Using the formula:

P(obtain rth success on kth trial) = k-1Cr-1 * pr * (1-p)k – r

P(meet the 3rd left-handed student on the 10th student) = 10-1C3-1 * (.1)3 * (1-.1)10 – 3

P(obtain second make on fifth shot) = (36) * (.001) * (0.4782969) = .01722

Using the calculator:

Plug the following numbers into the Negative Binomial Distribution Calculator:


The probability that the 10th student we survey is the third left-handed student we encounter is .01722.

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