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What Is Hypergeometric Distribution?

By David Isaac Rudel
Updated May 23, 2024
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Hypergeometric distribution describes the probability of certain events when a sequence of items is drawn from a fixed set, such as choosing playing cards from a deck. The key characteristic of events following the hypergeometric probability distribution is that the items are not replaced between draws. After a particular object has been chosen, it cannot be chosen again. This feature is most significant when working with small populations.

Quality assessment auditors use the hypergeometric distribution when analyzing the number of defective products in a given group. Products are set aside after being tested because there is no reason to test the same product twice. Thus, the selection is done without replacement.

Poker probabilities are calculated using the hypergeometric distribution because cards are not shuffled back into the deck within a given hand. Initially, for example, one-fourth of the cards in a standard deck are spades, but the likelihood of being dealt two cards and finding both of them to be spades is not 1/4 * 1/4 = 1/16. After receiving the first spade, there are fewer spades left in the deck, so the probability of being dealt another spade is only 12/51. Hence, the probability of being dealt two cards and finding them both to be spades is 1/4 * 12/51 = 1/17.

Objects are not replaced between draws, so the probability of extreme scenarios is reduced for a hypergeometric distribution. One can compare being dealt red or black cards from a standard deck to flipping a coin. A fair coin will land on “heads” half the time, and half the cards in a standard deck are black. Yet the likelihood of getting five consecutive heads when flipping a coin is greater than the likelihood of being dealt a five-card hand and finding them all to be black cards. The probability of five consecutive heads is 1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 1/32, or about 3 percent, and the likelihood of five black cards is 26/52 * 25/51 * 24/50 * 23/49 * 22/48 = 253/9996, or about 2.5 percent.

Sampling without replacement reduces the likelihood of extreme cases, but it does not affect the arithmetic mean of the distribution. The average number of heads expected when one flips a coin five times is 2.5, and this equals the average number of black cards expected in a five-card hand. Just as it is very unlikely that all five cards are black, it also is unlikely that none of them are. This is described in mathematical language by saying that replacement lowers the variance without affecting the expected value of a distribution.

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