Terms you'll find helpful in completing today's challenge are outlined below.

Negative Binomial Experiment

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

• The experiment consists of repeated trials.
• The trials are independent.
• The outcome of each trial is either success () or failure ().
• is the same for every trial.
• The experiment continues until successes are observed.
  
  

If is the number of experiments until the success occurs, then is a discrete random variable called a negative binomial.

Negative Binomial Distribution

Consider the following probability mass function:

The function above is negative binomial and has the following properties:

• The number of successes to be observed is .
• The total number of trials is .
• The probability of success of trial is .
• The probability of failure of trial , where .
• is the negative binomial probability, meaning the probability of having successes after trials and having successes after trials.
  
  

Note: Recall that . For further review, see the Combinations and Permutations Tutorial.

Geometric Distribution

The geometric distribution is a special case of the negative binomial distribution that deals with the number of Bernoulli trials required to get a success (i.e., counting the number of failures before the first success). Recall that is the number of successes in independent Bernoulli trials, so for each (where ):

The geometric distribution is a negative binomial distribution where the number of successes is . We express this with the following formula:

Example

Bob is a high school basketball player. He is a free throw shooter, meaning his probability of making a free throw is . What is the probability that Bob makes his first free throw on his fifth shot?

For this experiment, , and , So,

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