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An Introduction to the Hypergeometric Distribution

An introduction to the hypergeometric distribution. I briefly discuss the difference between sampling with replacement and sampling without replacement. I describe the conditions required for the hypergeometric distribution to hold, discuss the formula, and work through 2 simple examples. I also discuss the relationship between the binomial distribution and the hypergeometric distribution, and a rough guideline for when the binomial distribution can be used as a reasonable approximation to the hypergeometric. I finish with a brief example involving the multivariate hypergeometric distribution. For those using R, here is the R code to find the probabilities for the examples in this video: The probability of picking exactly 4 red balls when picking 5 balls from a source containing 6 red and 14 yellow. Without replacement (hypergeometric): choose(6,4)*choose(14,1)/choose(20,5) [1] 0.01354489 or dhyper(4,6,14,5) [1] 0.01354489 With replacement (binomial): dbinom(4,5,6/20) [1] 0.02835 The probability of picking exactly 7 females when randomly sampling from a school with 1100 female and 900 male students. Without replacement (hypergeometric): choose(1100,7)*choose(900,3)/choose(2000,10) [1] 0.1664901 or dhyper(7,1100,900,10) [1] 0.1664901 With replacement (binomial): dbinom(7,10,1100/2000) [1] 0.1664783 Multivariate hypergeometric, probability of picking exactly 3 Democrats, 2 Republicans, and 1 independent in the sample. choose(12,3)*choose(24,2)*choose(8,1)/choose(44,6) [1] 0.06881377 or, with the extraDistr package installed: dmvrhyper(c(3,2,1),c(12,24,8),6)