Hypergeometric probability

Definition

Hypergeometric probability is the probability of $k$ successes in $n$ draws from a finite population without replacement. It is used when the sample size and the population size are both known, and each draw changes the composition of the population.

5 Must Know Facts For Your Next Test

1. The hypergeometric distribution models scenarios where sampling is done without replacement.
2. The probability mass function (PMF) for hypergeometric distribution is given by $P(X=k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}$, where $N$ is the population size, $K$ is the number of successes in the population, $n$ is the sample size, and $k$ is the number of observed successes.
3. The mean of a hypergeometric distribution is given by $E(X) = n \frac{K}{N}$.
4. The variance of a hypergeometric distribution is given by $Var(X) = n \frac{K}{N} \left(1 - \frac{K}{N}\right) \left(\frac{N-n}{N-1}\right)$.
5. Hypergeometric probabilities can be calculated using combinatorial methods or statistical software.

Related terms

Binomial Distribution:

A probability distribution that describes the number of successes in a fixed number of independent trials with replacement, each with its own success probability.

Combinatorics:

A branch of mathematics dealing with combinations, permutations, and counting, often used in calculating probabilities involving discrete events.

Probability Mass Function (PMF): A function that gives the probability that a discrete random variable takes on a specific value.