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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 \(n\) ways for a "good" selection and \(m\) ways for a "bad" selection out of a total of \(n+m\) possibilities. Take \(N\) samples and let \(x_i\) equal \(1\) if selection \(i\) is successful and \(0\) if it is not. Let \(x\) be the total number of successful selections,


The probability of \(i\) 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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