The hypergeometric distribution calculates the probability of events without replacement, such as drawing cards from a deck. It is used in quality assessment and poker odds. Sampling without replacement reduces extreme cases but not the mean.
The 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 feature of events following the hypergeometric probability distribution is that items are not replaced between draws. Once a particular item has been chosen, it cannot be chosen again. This feature is more significant when working with small populations.
Quality assessment reviewers 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. Therefore, the selection is done without replacement.
Poker odds are calculated using the hypergeometric distribution because the cards are not shuffled back into the deck within any given hand. Initially, for example, a quarter of the cards in a standard deck are spades, but the probability that two cards are dealt and both are not spades is 1/4 * 1/4 = 1/16. After receiving the first spade, there are fewer spades left in the deck, so the probability of another spade being dealt is only 12/51. Thus, the probability of being dealt two cards and finding both spades is 1/4 * 12/51 = 1/17.
Objects are not replaced between draws, so the likelihood of extreme scenarios is reduced for a hypergeometric distribution. Dealing red or black cards from a standard deck can be likened to tossing a coin. A fair coin will land on heads half the time, and half the cards in a standard deck are black. However, the probability of getting five consecutive heads when flipping a coin is greater than the probability of being dealt a hand of five cards and finding all of them as 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 probability 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 probability of extreme cases, but does not affect the arithmetic mean of the distribution. The average number of heads expected when flipping a coin five times is 2.5, and this is equal to the average number of black cards expected in a hand of five cards. Just as it is very unlikely that all five cards are black, it is equally unlikely that any of them are. This is described in mathematical language by saying that substitution reduces the variance without affecting the expected value of a distribution.
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