Practice problems for these concepts can be found at:

- Discrete Random Variables Solved Problems for Beginning Statistics
- Discrete Random Variables Supplementary Problems for Beginning Statistics

### Hypergeometric Random Variable

The *hypergeometric random variable* is used in situations where success or failure is possible on each trial but where there is not independence from trial to trial. The lack of independence from trial to trial distinguishes the hypergeometric distribution from the binomial distribution. The hypergeometric random variable applies in situations where there are N items, of which k are classified as successes and N – k are classified as failures. A sample of size n ≤ k is selected from the N items and X is defined to equal the number of successes in the n items selected. X is a hypergeometric random variable which can equal any whole number from 0 to n.

**EXAMPLE 5.26** A sociologist randomly selects 5 individuals from a group consisting of 10 male single parents and 15 female single parents. The random variable X is defined to equal the number of male single parents in the 5 selected individuals. In this example, N = 25, k = 10, N – k = 15, and n = 5. The number of male single parents in the 5 selected is a hypergeometric random variable. If this hypergeometric random variable is represented by X, then X may assume any one of the values 0, 1, 2, 3, 4, or 5.

**EXAMPLE 5.27** A box contains 5 defective and 25 acceptable computer monitors. Three of the monitors are randomly selected and X is defined to be the number of defective monitors in the three. X is a hypergeometric random variable with N = 30, k = 5, N – k = 25, and n = 3. X may assume any one of the values 0, 1, 2, or 3.

### Hypergeometric Probability Formula

When n items are selected from N items of which k are successes and N – k are failures, the random variable X, defined to equal the number of successes in the n selected items, is a hypergeometric random variable. The probability distribution of X is given by the *hypergeometric probability formula* shown in formula (*5*.*13*).

**EXAMPLE 5.28** A police department consists of 25 officers of whom 5 are minorities. Three officers are randomly selected to meet with the mayor. Let X be the number of minorities in the three selected to meet with the mayor. X is a hypergeometric random variable with N = 25, k = 5, N – k = 20, and n = 3. The probability distribution of X is derived as follows:

The EXCEL output for the hypergeometric distribution is given in Table 5.10. The probability that X ≤ 1 is given by adding P(0) and P(1) to get 0.909.

**EXAMPLE 5.29** The binomial distribution approximates the hypergeometric distribution whenever n ≤ .05N. A box contains 200 computer chips, of which 7 are defective. The probability of finding one defective in a sample of 5 randomly selected chips is given by the following hypergeometric probability computation.

Since n ≤ .05 × 200 = 10, the probability may be approximated by using the binomial distribution. The five selections of the computer chips may be viewed as n = 5 trials. The probability of success, selecting a defective chip, is p = = .035, and q = .965. The binomial probability of one defective in the five chips is given as follows:

The approximation is very good when n ≤ .05N, as shown in this example.

Practice problems for these concepts can be found at:

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