# Negative Binomial Distribution Calculator

Use the Negative Binomial Calculator to compute probabilities, given a negative binomial experiment. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems.

• Enter a value in each of the first three text boxes (the unshaded boxes).
• Click the Calculate button.
• The Calculator will compute the Negative Binomial Probability.
 Number of trials Number of successes Probability of success on a trial Negative binomial probability

Instructions: To find the answer to a frequently-asked question, simply click on the question. If none of the questions addresses your need, refer to Stat Trek's tutorial on the negative binomial distribution or visit the Statistics Glossary.

### What is a negative binomial experiment?

A negative binomial experiment is a statistical experiment that has the following properties:

• The experiment consists of x repeated trials.
• Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
• The probability of success, denoted by p, is the same on every trial.
• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.
• The experiment continues until r successes are observed, where r is specified in advance.

Consider the following statistical experiment. You flip a coin repeatedly and count the number of times the coin lands on heads. You continue flipping the coin until it has landed 5 times on heads. This is a negative binomial experiment because:

• The experiment consists of repeated trials. We flip a coin repeatedly until it has landed 5 times on heads.
• Each trial can result in just two possible outcomes - heads or tails.
• The probability of success is constant - 0.5 on every trial.
• The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.
• The experiment continues until a fixed number of successes have occurred; in this case, 5 heads.

### What is a negative binomial distribution?

The probability distribution of a negative binomial random variable is called a negative binomial distribution. The negative binomial distribution is also known as the Pascal 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. The negative binomial random variable is the number of coin flips required to achieve 2 heads. In this example, the number of coin flips is a random variable that can take on any integer value between 2 and plus infinity. The negative binomial probability distribution for this example is presented below.

Number of coin flips Probability
2 0.25
3 0.25
4 0.1875
5 0.125
6 0.078125
7 or more 0.109375

### What is the number of trials?

The number of trials refers to the number of attempts in a negative binomial experiment.

Suppose that we conduct the following negative binomial experiment. We flip a coin and count the number of flips until the coin has landed three times on Heads. If we need to flip the coin 5 times until the coin has landed on Heads 3 times, then 5 is the number of trials.

### What is the number of successes?

Each trial in a negative binomial experiment can have one of two outcomes. The experimenter classifies one outcome as a success; and the other, as a failure. The number of successes in a binomial experient is the number of trials that result in an outcome classified as a success.

### What is the probability of success on a trial?

In a negative binomial experiment, the probability of success on any individual trial is constant. For example, the probability of getting Heads on a single coin flip is always 0.50. If "getting Heads" is defined as success, the probability of success on a single trial would be 0.50.

### What is the negative binomial probability?

The negative binomial probability refers to the probability that a negative binomial experiment results in r - 1 successes after trial x - 1 and r successes after trial x.

For example, suppose we conduct a negative binomial experiment to count the number of coin flips required for a coin to land 2 times on Heads. We might ask: What is the probability that this experiment will require 5 coin flips? In this example, we would be asking about a negative binomial probability. (If you use the Negative Binomial Calculator to analyze this experiment, you will find that the probability that this experiment would require 5 coin flips is 0.125.)

### How is a binomial experiment related to a negative binomial experiment?

A binomial experiment and a negative binomial experiment have exactly the same properties, except for one thing.

With a binomial experiment, we are concerned with finding the probability of r successes in x trials, where x is fixed. With a negative binomial experiment, we are concerned with finding the probability that the rth success occurs on the xth trial, where r is fixed.

### What is a geometric distribution related to a negative binomialdistribution?

The geometric distribution is a special case of the negative binomial distribution. It deals with the number of trials required for a single success. Thus, the geometric distribution is negative binomial distribution where the number of successes (r) is equal to 1.

With a negative binomial distribution, we are concerned with finding the probability that the rth success occurs on the xth trial, where r is fixed. With a geometric distribution, we are concerned with finding the probability that the first success occurs on the xth trial.

### Can I use the Negative Binomial Calculator to solve problems based on the geometric distribution?

Of course! The geometric distribution is just a special case of the negative binomial distribution (see above question); so geometric distribution problems can be solved with the Negative Binomial Calculator.

## Sample Problems

1. The probability that a driver passes the written test for a driver's license is 0.75. What is the probability that a person will fail the test on the first try and pass the test on the second try?

Solution:

We know the following:

• The number of trials is 2.
• The number of successes is 1 (since we define passing the test as success).
• The probability of success (i.e., passing the test) on any single trial is 0.75.

Therefore, we plug those numbers into the Negative Binomial Calculator and hit the Calculate button. The calculator reports that the negative binomial probability is 0.1875. That is the probability of failing the first test and passing the second test.

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

Solution:

We know the following:

• The number of trials is 9 (because we flip the coin nine times).
• The number of successes is 4 (since we define Heads as a success).
• The probability of success for any coin flip is 0.5.

Therefore, we plug those numbers into the Negative Binomial Calculator and hit the Calculate button. The calculator reports that the negative binomial probability is 0.109375. That is the probability that the coin will land on heads for the fourth time on the ninth coin flip.