Table of Contents
The Hypergeometric Distribution is a probability distribution that describes the probability of obtaining a specific number of successes in a fixed number of draws from a finite population, without replacement. It differs from other probability distributions, such as the Binomial and Poisson distributions, in that it takes into account the size of the population and the number of successes in the population, rather than assuming an infinite population and a fixed probability of success. This makes it particularly useful in situations where the sample size is small and the population size is known. Additionally, the Hypergeometric Distribution is non-symmetrical and is often used in situations where the sample size is a small proportion of the population size.
An Introduction to the Hypergeometric Distribution
The hypergeometric distribution describes the probability of choosing k objects with a certain feature in n draws without replacement, from a finite population of size N that contains K objects with that feature.
If a X follows a hypergeometric distribution, then the probability of choosing k objects with a certain feature can be found by the following formula:
P(X=k) = KCk (N-KCn-k) / NCn
where:
- N: population size
- K: number of objects in population with a certain feature
- n: sample size
- k: number of objects in sample with a certain feature
- KCk: number of combinations of K things taken k at a time
For example, there are 4 Queens in a standard deck of 52 cards. Suppose we randomly pick a card from a deck, then, without replacement, randomly pick another card from the deck. What is the probability that both cards are Queens?
To answer this, we can use the hypergeometric distribution with the following parameters:
- N: population size = 52 cards
- K: number of objects in population with a certain feature = 4 queens
- n: sample size = 2 draws
- k: number of objects in sample with a certain feature = 2 queens
Plugging these numbers in the formula, we find the probability to be:
P(X=2) = KCk (N-KCn-k) / NCn = 4C2 (52-4C2-2) / 52C2 = 6*1/ 1326 = 0.00452.
This should make sense intuitively. If you imagine yourself pulling two cards out of a deck, one after the other, the probability that both cards are Queens should be very low.
Properties of the Hypergeometric Distribution
The hypergeometric distribution has the following properties:
The mean of the distribution is (nK) / N
The variance of the distribution is (nK)(N-K)(N-n) / (N2(n-1))
Hypergeometric Distribution Practice Problems
Use the following practice problems to test your knowledge of the hypergeometric distribution.
Problem 1
Question: Suppose we randomly pick four cards from a deck without replacement. What is the probability that two of the cards are Queens?
To answer this, we can use the hypergeometric distribution with the following parameters:
- N: population size = 52 cards
- K: number of objects in population with a certain feature = 4 queens
- n: sample size = 4 draws
- k: number of objects in sample with a certain feature = 2 queens
Plugging these numbers into the Hypergeometric Distribution Calculator, we find the probability to be 0.025.
Problem 2
Question: An urn contains 3 red balls and 5 green balls. You randomly choose 4 balls. What is the probability that you choose exactly 2 red balls?
To answer this, we can use the hypergeometric distribution with the following parameters:
- N: population size = 8 balls
- K: number of objects in population with a certain feature = 3 red balls
- n: sample size = 4 draws
- k: number of objects in sample with a certain feature = 2 red balls
Plugging these numbers into the Hypergeometric Distribution Calculator, we find the probability to be 0.42857.
Problem 3
Question: A basket contains 7 purple marbles and 3 pink marbles. You randomly choose 6 marbles. What is the probability that you choose exactly 3 pink marbles?
To answer this, we can use the hypergeometric distribution with the following parameters:
- N: population size = 10 marbles
- K: number of objects in population with a certain feature = 3 pink balls
- n: sample size = 6 draws
- k: number of objects in sample with a certain feature = 3 pink balls
Plugging these numbers into the Hypergeometric Distribution Calculator, we find the probability to be 0.16667.