The hypergeometric distribution describes the probability of selecting exactly $x$ items *from a sub-population* of size $K$, when drawing $n$ times *without replacement* from a population of size $N$. It models situations where the total population and sub-population are finite and fixed, and draws are dependent on previous outcomes. | Parameter | Description | | --------- | ------------------------------------------- | | $N$ | Size of the overall population | | $K$ | Size of the sub-population | | $n$ | Number of draws | | $x$ | Number of draws belonging to sub-population | ## Probability Mass Function The probability of observing exactly $x$ successes (items from the sub-population) is given by the following [[Probability Mass Function|PMF]]: $ \mathbf P(X=x)=\frac{\begin{pmatrix} K\\x \end{pmatrix} * \begin{pmatrix} N-K\\n-x \end{pmatrix}} { \begin{pmatrix} N\\n \end{pmatrix}} $ **Numerator:** - *First term:* Number of ways to choose $x$ items from the sub-population $K$. - *Second term:* Number of ways to choose the remaining $(n-x)$ draws from outside the sub-population $(N-K)$. **Denominator:** Number of ways to choose $n$ items from a population of $N$. ![[hypergeometric-distribution.png|center|400]] ## Comparison to Multinomial Distribution | | Hypergeometric | [[Multinomial Distribution]] | | | --------------- | -------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------- | --- | | Sampling Method | Draws are without replacement. | Draws are with replacement or modeled as [[Independence of Events\|independent]] independent trials. | | | Independence | Probabilities of subsequent draws depend on earlier outcomes, as the total population size decreases with each draw. | Probabilities remain constant across all trials, as each draw does not affect the outcomes of others. | | | Example | If a red ball is drawn, there is one less red ball available for subsequent draws, altering the probabilities. | Rolling a die multiple times assumes each roll has the same probability distribution, independent of previous outcomes. | |