The negative binomial distribution describes the probability of experiencing a certain amount of failures before experiencing a certain amount of successes in a series of Bernoulli trials.
A Bernoulli trial is an experiment with only two possible outcomes – “success” or “failure” – and the probability of success is the same each time the experiment is conducted.
An example of a Bernoulli trial is a coin flip. The coin can only land on two sides (we could call heads a “success” and tails a “failure”) and the probability of success on each flip is 0.5, assuming the coin is fair.
If a random variable X follows a negative binomial distribution, then the probability of experiencing k failures before experiencing a total of r successes can be found by the following formula:
P(X=k) = k+r-1Ck * (1-p)r *pk
where:
- k: number of failures
- r: number of successes
- p: probability of success on a given trial
- k+r-1Ck: number of combinations of (k+r-1) things taken k at a time
For example, suppose we flip a coin and define a “successful” event as landing on heads. What is the probability of experiencing 6 failures before experiencing a total of 4 successes?
To answer this, we can use the negative binomial distribution with the following parameters:
- k: number of failures = 6
- r: number of successes = 4
- p: probability of success on a given trial = 0.5
Plugging these numbers in the formula, we find the probability to be:
P(X=6 failures) = 6+4-1C6 * (1-.5)4 *(.5)6 = (84)*(.0625)*(.015625) = 0.08203.
Properties of the Negative Binomial Distribution
The negative binomial distribution has the following properties:
The mean number of failures we expect before achieving r successes is pr / (1-p).
The variance in the number of failures we expect before achieving r successes is pr / (1-p)2.
For example, suppose we flip a coin and define a “successful” event as landing on heads.
The mean number of failures (e.g. landing on tails) we expect before achieving 4 successes would be pr/(1-p) = (.5*4) / (1-.5) = 4.
The variance in the number of failures we expect before achieving 4 successes would be pr / (1-p)2 = (.5*4) / (1-.5)2 = 8.
Negative Binomial Distribution Practice Problems
Use the following practice problems to test your knowledge of the negative binomial distribution.
Note: We will use the Negative Binomial Distribution Calculator to calculate the answers to these questions.
Problem 1
Question: Suppose we flip a coin and define a “successful” event as landing on heads. What is the probability of experiencing 3 failures before experiencing a total of 4 successes?
Answer: Using the Negative Binomial Distribution Calculator with k = 3 failures, r = 4 successes, and p = 0.5, we find that P(X=3) = 0.15625.
Problem 2
Question: Suppose we go door to door selling candy. We consider it a “success” if someone buys a candy bar. The probability that any given person will buy a candy bar is 0.4. What is the probability of experiencing 8 failures before we experience a total of 5 successes?
Answer: Using the Negative Binomial Distribution Calculator with k = 8 failures, r = 5 successes, and p = 0.4, we find that P(X=8) = 0.08514.
Problem 3
Question: Suppose we roll a die and define a “successful” roll as landing on the number 5. The probability that the die lands on a 5 on any given roll is 1/6 = 0.167. What is the probability of experiencing 4 failures before we experience a total of 3 successes?
Answer: Using the Negative Binomial Distribution Calculator with k = 4 failures, r = 3 successes, and p = 0.167, we find that P(X=4) = 0.03364.