Hypergeometric Distribution: Probability Without Replacement

Understand the hypergeometric distribution and its application in calculating probabilities when sampling without replacement from a finite population. This guide explains the hypergeometric probability formula and provides examples to illustrate its use.

Hypergeometric Distribution in Discrete Mathematics

Understanding the Hypergeometric Distribution

The hypergeometric distribution is used to calculate probabilities when you're selecting items from a finite population without replacement. Imagine you have a bag of marbles, some red and some blue. You reach in and grab a handful. The probability of getting a certain number of red marbles depends on the total number of marbles, the number of red marbles, and how many you grab.

Key Parameters of the Hypergeometric Distribution

  • N: The total number of items in the population.
  • K: The number of "successes" (items of interest) in the population.
  • n: The number of items selected (sample size).
  • x: The number of successes observed in the sample.

The Hypergeometric Formula

The probability of getting exactly x successes in a sample of size n is given by:

P(X = x) = [C(K, x) * C(N - K, n - x)] / C(N, n)

Where C(a, b) is the number of combinations of choosing b items from a set of a items (a! / (b! * (a-b)!)).

Relationship to the Binomial Distribution

The hypergeometric distribution is similar to the binomial distribution, which is used for sampling with replacement or when the population size is much larger than the sample size. The binomial distribution can approximate the hypergeometric distribution under these conditions (when the sample size is small relative to the population size).

Examples of the Hypergeometric Distribution

Example 1: Drawing Cards

You draw 5 cards from a standard deck of 52 cards (without replacement). What's the probability of getting exactly 2 red cards?

  • N = 52
  • K = 26 (number of red cards)
  • n = 5
  • x = 2

P(X = 2) = [C(26, 2) * C(26, 3)] / C(52, 5) ≈ 0.325

Example 2: Drawing Balls

A bag contains 21 balls (13 orange, 8 green). You draw 5 balls (without replacement). What's the probability of getting exactly 3 orange balls?

  • N = 21
  • K = 13
  • n = 5
  • x = 3

P(X = 3) = [C(13, 3) * C(8, 2)] / C(21, 5)

Properties of the Hypergeometric Distribution

Key statistical measures for the hypergeometric distribution include:

  • Mean (Expected Value): The average number of successes.
  • Median: The middle value.
  • Mode: The most frequent value.
  • Variance: A measure of the spread of the data.

Conclusion

The hypergeometric distribution is a valuable tool for calculating probabilities when sampling without replacement. Its application extends to many areas requiring the precise calculation of probabilities in finite populations.