A **negative binomial distribution** tells us how many trials (*k*) are required until we obtain the r^{th} “success.”

Given that the probability of success in a given trial equals *p*, we can find the probability that we will obtain the r^{th} success on the k^{th} trial using the formula:

P(obtain r^{th} success on k^{th} trial) = _{k-1}C_{r-1} * p^{r} * (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 *r *= 2^{nd }shot on his *k *= 5^{th} attempt.

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

**Using the formula:**

P(obtain r^{th} success on k^{th} trial) = _{k-1}C_{r-1} * p^{r} * (1-p)^{k – r}

P(obtain second make on fifth shot) = _{5-1}C_{2-1} * (.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 *k *= 10^{th} student is the *r *= 3^{rd }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 r^{th} success on k^{th} trial) = _{k-1}C_{r-1} * p^{r} * (1-p)^{k – r}

P(meet the 3^{rd }left-handed student on the 10^{th} student) = _{10-1}C_{3-1} * (.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**.