Hypergeometric distributions: A very brief introduction.

This will be a very abstract introduction, so you'll have to get used to it, and then you'll have to extend this abstraction to actual applications in the problems that are collected in this set.

The basic idea here is that we're trying to calculate the probability of getting a fixed number of "good" elements out of "bad" elements by picking a specified number of arbitrary elements. Those elements can be balls, atoms, numbers...

Let there be nn ways for a "good" selection and mm ways for a "bad" selection out of a total of n+mn+m possibilities. Take NN samples and let xix_i equal 11 if selection ii is successful and 00 if it is not. Let xx be the total number of successful selections,


The probability of ii successful selections is then

\eqalign{P(x=i) &= \dfrac{[\#\text{ ways for }i\text{ successes}][\#\text{ ways for }N-i\text{ failures}]}{[\text{total number of ways to select}]} \\ &= \dfrac{\dbinom ni\dbinom m{N-i}}{\dbinom{m+n} N}\\ &= \dfrac{m!n!N!(m+n-N)!}{i!(n-i)!(m+i-N)!(N-i)!(m+n)!}.} You will need this last formula to solve most of the problems.

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