Binomial Approximation to a Hypergeometric Random Variable

sample size n

20

number of marked balls k

25

total number of balls N

100

If

n

balls are sampled from a bin containing

N

balls of which

k

are marked, then the distribution of the number of marked balls in the sample depends on whether the sampling is done with or without replacement.

If the sampling is done with replacement, the number of marked balls is a binomial random variable, while if the sampling is done without replacement, then this quantity is a hypergeometric random variable. However, if the number of balls sampled is small relative to the number of balls in the bin, it makes little difference how the sampling is done.

The probability histogram for the binomial random variable is colored orange, and the hypergeometric random variable is blue. A measure of agreement between the two is obtained by computing the purple area—100% would represent complete agreement between the two distributions.