Hypergeometric Probability Distribution

Hypergeometric Probability Distribution

Learning Objectives

Determine whether a probability experiment is a hypergeometric experiment

Compute the probabilities of hypergeometric experiments

Compute the mean and the variance/standard deviation of a hypergeometric random variable

Recall: Binomial Probability

The binomial probability distribution can be used to compute the probabilities of experiments when there are a fixed number of trials in which there are two mutually exclusive outcomes and the probability of success for any trial is constant.

What if the requirement of independence is not

satisfied?

Criteria for a Hypergeometric Probability Experiment

A probability experiment is said to be a hypergeometric experiment provided:1. The finite population to be sampled has N

elements.2. For each trial of the experiment, there are

two possible outcomes, success or failure. There are exactly k successes in the population.

3. A sample of size n is obtained from the population of size N without replacement.

Notations Used in the Hypergeometric Probability Distribution

The population is size N. The sample is size n.

There are k successes in the population.Let the random variable X denote the

number of successes in the sample of size n, so x must be greater than or equal to the larger of 0 or n – (N – k), and x must be less than or equal to the smaller of n or k.

Example 1:A Hypergeometric Probability Experiment

Problem:

Suppose that a researcher goes to a small college with 200 faculty, 12 of which have blood type O-negative. She obtains a simple random sample of n = 20 of the faculty and finds that 3 of the faculty have blood type O-negative. Is this experiment a hypergeometric probability experiment? List the possible values of the random variable X, the number of faculty that have blood type O-negative.

Example 1:A Hypergeometric Probability Experiment

Approach:

We need to determine if the three criteria for a hypergeometric experiment have been satisfied.

Example 1:A Hypergeometric Probability Experiment

Solution:This is a hypergeometric probability experiment because1. The population consists of N = 200 faculty.2. Two outcomes are possible: the faculty member has

blood type O-negative or the faculty member does not have blood type O-negative. The researcher obtained k = 3 successes.

3. The sample size n = 20.

The possible values of the random variable are x = 0, 1, …, 12. The largest value of X is 12, because we cannot have more than 12 successes since there are only 12 faculty with blood type O-negative in the population.

Notice that we cannot use the binomial probability distribution to determine the likelihood of obtaining three successes in 20 trials in Example 1 because the sample size is large relative to the population size.

That is, n = 20 is more than 5% of the population size, N = 200.

The basis for computing probabilities in a hypergeometric experiment lies in the fact that each sample of size n is equally likely to be chosen.

Consider an urn that contains 8 white chips and 6 black chips for a total of N = 14 chips. If we decide to randomly select n = 3, all possible combinations of chips are equally likely.

That is, if we let W1,W2, …, W8 represent the 8 white chips and B1, B2, …, B6 represent the 6 black chips, selecting W1,W2, B3 is just as likely as selecting W3,W6, B4.

Notice in both cases that we selected 2 white chips and 1 black chip. So, if X represents the number of black chips selected, we have x = 1 in both cases; however, the chips selected are different (so each represents a different sample).

Hypergeometric Probability Distribution

The probability of obtaining x successes based on a random sample of size n from a population of size N is given by

where k is the number of successes in the population.

Example 2Using the Hypergeometric Probability

Distribution

Problem:

Suppose that a researcher goes to a small college with 200 faculty, 12 of which have blood type O-negative. She obtains a simple random sample of n = 20 of the faculty. Let the random variable X represent the number of faculty in the sample size of n = 20 that have blood type O-negative.

a) What is the probability that 3 of the faculty have blood type O-negative?

b) What is the probability that at least one of the faculty has blood type O-negative?

Example 2Using the Hypergeometric Probability

Distribution

Approach:

This is a hypergeometric experiment with N = 200, n = 20, and k = 12. The possible values of the random variable X are x = 0, 1, 2, …, 12.

Example 2Using the Hypergeometric Probability

Distribution

Solution:a) We are looking for the probability of

obtaining 3 successes, x = 3.

Example 2Using the Hypergeometric Probability

Distribution

Solution:b) Since ,

Try this!

Suppose that a machine shop orders 500 bolts from a supplier. To determine whether to accept the shipment of bolts, the manager of the facility randomly selects 12 bolts. If none of the 12 randomly selected bolts is found to be defective, he concludes that the shipment is acceptable.a) If 10% of the bolts in the population are

defective, what is the probability that none of the selected bolts are defective?

b) If 20% of the bolts in the population are defective, what is the probability that none of the selected bolts are defective?

Mean and Standard Deviation of a Hypergeometric Random Variable

A hypergeometric random variable X has mean and standard deviation given by the formulas

where n is the sample sizek is the number of successes in the

populationN is the size of the population

Computing the Mean and Standard Deviation

of a Hypergeometric Random Variable