The hypergeometric distribution is a discrete distribution that models the number of events in a fixed sample size when you know the total number of items in the population that the sample is from. Each item in the sample has two possible outcomes (either an event or a nonevent). The samples are without replacement, so every item in the sample is different. When an item is chosen from the population, it cannot be chosen again. Therefore, an item's chance of being selected increases on each trial, assuming that it has not yet been selected.
Use the hypergeometric distribution for samples that are drawn from relatively small populations, without replacement. For example, the hypergeometric distribution is used in Fisher's exact test to test the difference between two proportions, and in acceptance sampling by attributes for sampling from an isolated lot of finite size.
The hypergeometric distribution is defined by 3 parameters: population size, event count in population, and sample size.
For example, you receive one special order shipment of 500 labels. Suppose that 2% of the labels are defective. The event count in the population is 10 (0.02 * 500). You sample 40 labels and want to determine the probability of 3 or more defective labels in that sample. The probability of 3 of more defective labels in the sample is 0.0384.
The probability distribution of a hypergeometric random variable is called a hypergeometric distribution. This lesson describes how hypergeometric random variables, hypergeometric experiments, hypergeometric probability, and the hypergeometric distribution are all related.
Notation
The following notation is helpful, when we talk about hypergeometric distributions and hypergeometric probability.
- h(x; N, n, k): hypergeometric probability - the probability that an n-trial hypergeometric experiment results in exactly x successes, when the population consists of N items, k of which are classified as successes.
Hypergeometric Experiments
A hypergeometric experiment is a statistical experiment that has the following properties:
- In the population, k items can be classified as successes, and N - k items can be classified as failures.
Consider the following statistical experiment. You have an urn of 10 marbles - 5 red and 5 green. You randomly select 2 marbles without replacement and count the number of red marbles you have selected. This would be a hypergeometric experiment.
Note that it would not be a binomial experiment. A binomial experiment requires that the probability of success be constant on every trial. With the above experiment, the probability of a success changes on every trial. In the beginning, the probability of selecting a red marble is 5/10. If you select a red marble on the first trial, the probability of selecting a red marble on the second trial is 4/9. And if you select a green marble on the first trial, the probability of selecting a red marble on the second trial is 5/9.
Note further that if you selected the marbles with replacement, the probability of success would not change. It would be 5/10 on every trial. Then, this would be a binomial experiment.
Hypergeometric Distribution
A hypergeometric random variable is the number of successes that result from a hypergeometric experiment. The probability distribution of a hypergeometric random variable is called a hypergeometric distribution.
Given x, N, n, and k, we can compute the hypergeometric probability based on the following formula:
Hypergeometric Formula.. Suppose a population consists of N items, k of which are successes. And a random sample drawn from that population consists of n items, x of which are successes. Then the hypergeometric probability is:
h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]
The hypergeometric distribution has the following properties:
- The variance is n * k * ( N - k ) * ( N - n ) / [ N2 * ( N - 1 ) ] .
Example 1
Suppose we randomly select 5 cards without replacement from an ordinary deck of playing cards. What is the probability of getting exactly 2 red cards (i.e., hearts or diamonds)?
Solution: This is a hypergeometric experiment in which we know the following:
- x = 2; since 2 of the cards we select are red.
We plug these values into the hypergeometric formula as follows:
h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]
h(2; 52, 5, 26) = [ 26C2 ] [ 26C3 ] / [ 52C5 ]
h(2; 52, 5, 26) = [ 325 ] [ 2600 ] / [ 2,598,960 ]
h(2; 52, 5, 26) = 0.32513
Thus, the probability of randomly selecting 2 red cards is 0.32513.
Cumulative Hypergeometric Probability
A cumulative hypergeometric probability refers to the probability that the hypergeometric random variable is greater than or equal to some specified lower limit and less than or equal to some specified upper limit.
For example, suppose we randomly select five cards from an ordinary deck of playing cards. We might be interested in the cumulative hypergeometric probability of obtaining 2 or fewer hearts. This would be the probability of obtaining 0 hearts plus the probability of obtaining 1 heart plus the probability of obtaining 2 hearts, as shown in the example below.
Example 2
Suppose we select 5 cards from an ordinary deck of playing cards. What is the probability of obtaining 2 or fewer hearts?
Solution: This is a hypergeometric experiment in which we know the following:
- x = 0 to 2; since our selection includes 0, 1, or 2 hearts.
We plug these values into the hypergeometric formula as follows:
h(x < x; N, n, k) = h(x < 2; 52, 5, 13)
h(x < 2; 52, 5, 13) = h(x = 0; 52, 5, 13) + h(x = 1; 52, 5, 13) + h(x = 2; 52, 5, 13)
h(x < 2; 52, 5, 13) = [ (13C0) (39C5) / (52C5) ] + [ (13C1) (39C4) / (52C5) ] + [ (13C2) (39C3) / (52C5) ]
h(x < 2; 52, 5, 13) = [ (1)(575,757)/(2,598,960) ] + [ (13)(82,251)/(2,598,960) ] + [ (78)(9139)/(2,598,960) ]
h(x < 2; 52, 5, 13) = [ 0.2215 ] + [ 0.4114 ] + [ 0.2743 ]
h(x < 2; 52, 5, 13) = 0.9072
Thus, the probability of randomly selecting at most 2 hearts is 0.9072.
Hypergeometric Calculator
As you surely noticed, the hypergeometric formula requires many time-consuming computations. The Stat Trek Hypergeometric Calculator can do this work for you - quickly, easily, and error-free. Use the Hypergeometric Calculator to compute hypergeometric probabilities and cumulative hypergeometric probabilities. The calculator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below.
Hypergeometric Calculator