> [!note] Hypergeometric Distribution > $ > P(X = k) = \frac{{m \choose k} {{n - m} \choose {r - k}}}{n \choose r} > $ > The hypergeometric distribution is a [[Probability Distribution|probability distribution]] of the hypergeometric [[Random Variable|random variable]], which represents the number of successes $k$ in $p$ draws, from a finite [[Population|population]] of size $n$ and $m$ successes, *without replacement*. > [!theoremb] Derivation > > The number of ways of [[Combination|choosing]] $k$ successes and $(r - k)$ failures is their product: > $ > {m \choose k}{n - m \choose r - k} > $ > Since there are $n \choose r$ ways of choosing the codeword, by the principle of symmetry, $P(X = k) = \frac{{m \choose k} {{n - m} \choose {r - k}}}{n \choose r}$.