Hypergeometric Distribution

Formulas

Suppose a finite population or batch has items that are either conforming or nonconforming. Let us define a nonconforming item to be a “success.” The probability distribution of the number of nonconforming items in the sample, denoted by X, is hypergeometric and is given as follows:

where

• D is the number of nonconforming items in the population,
• N is the size of the population,
• n is the size of the sample,
• x is the number of nonconforming items in the sample, and

• (Dx) is the combination of D items taken x at a time and is given by

  
• where the factorial of a positive integer x is written as x! and is given by x(x − 1)(x − 2)… 3.2.1.
• Further, 0! is defined to be 1.

The mean or expected value of a hypergeometric random variable is expressed as

The variance, a measure of dispersion, of a hyper-geometric random variable is given by

Example

A batch of 20 DVDs includes, unknown to the company, 3 nonconforming ones. If an inspector randomly samples 5 DVDs, what is the probability of getting 2 nonconforming DVDs? What are the mean and variance of the number of nonconforming DVDs in the sample?

Using the previously defined notation, and assuming that finding a nonconforming DVD is a “success,” we have N = 20, D = 3, n = 5, and x = 2:

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The complete probability distribution of X, the number of nonconforming DVDs in the sample, may be found using the formula, described previously, for values of X being 0, 1, 2, and 3. The results are shown below:

Note that the sum of all the probabilities equals 1.

Suppose an acceptance sampling plan calls for choosing a random sample of 5 from the batch of 20 DVDs. The batch is to be accepted if no nonconforming DVDs are found in the sample. To determine the chance of accepting the batch, as described previously, we use the calculated probability distribution values. Here, since the batch is accepted if the value of X = 0, the acceptance probability of the batch is 0.3991.

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