The Hypergeometric Distribution

This is a distribution you can impress your friend’s with because it has such an absolutely awesome sounding name and a comparatively simple form.

While the binomial distribution tells us about independent events the hypergeometric distribution tells us about a particular kind of dependent event, sampling without replacement. If you reach into a bag and take out a marble then reach in and take out another marble you are sampling without replacement. Like the binomial and Bernoulli distributions the hypergeometric describes events in terms of success and failure.

Here is our scenario in mathematical terms:

Randomly select n items from a set of N items. In this set there are K possible successes and N-K possible failures (which simply means everything is either a success of a failure). The probability of picking k successes from the set of K will follow the hypergeometric distribution. Which has this form:

Why? Let’s take a look.

Here is a random set from which we will pick three items. As per ancient statistical tradition consider it an urn containing while and black balls.

WBBBWBWBBWBBW

We will pick four balls from the urn and want to know how likely it is that exactly three of them will be black?

Here are our variables.

N = 13 # There are 13 items
K = 8  # 8 of them are black
n = 4  # We are selecting four items
k = 3  # Looking for 3 black

First we will describe how may ways there are to choose 3 black balls from a set of 8, because there are only 8 black balls in the set. Choose? That sounds like a binomial coefficient so let’s write _KC_k. Now how many ways are there to draw 1 white ball from a let of 5 white balls? That’s another binomial coefficient! We write it as _N-KC_n-k because N-K is the the number of non-black balls and n-k is the number of times we can pick a non-black ball.

Now we can simply multiply these together to get the number of correct combinations! There are 280 possible ways.

That’s not a probability, though, so we need to normalize it. How to do that? Well, how many ways are there to pick a combination from the set? That must be _NC_n since it is a set of N and we are choosing n items in total. That comes to 715.

So there are 280 correct results out of 715 total options. That’s just a fraction!

208/715
 0.3916084

There is a roughly 29% chance of us picking the particular combination of black and white balls that we want. And since we know that 280 is reakky _KC_k times _N-KC_n-k and that 715 is really _NC_n we can see where the equation comes from.

We can also summon up the hypergeometric distribution in R with the dhyper() family of functions.

dhyper(3,8,5,4)
 0.3916084

With so many variables there are tons of hypergeometric distributions but here’s a simple one that show the probability for between zero and four balls black chosen from an urn.

Successes <- 0:4
plot(dhyper(Successes,8,5,4)~Successes,
	type='h', lwd=3,
	ylim=c(0,.5), ylab='Probability',
	main='A Hypergeomatric Distribution')

Obviously the hypergeometric distribution is good for more than placing bets on Jacob Bernoulli’s less interesting party games or we wouldn’t be talking about it. The hypergeometric can be used to count cards. Perhaps most famously, however, it was used by Sir Ronald Fisher to devise a statistical test. He was given the task of testing the claim that fellow biologist Muriel Bristol-Roach could tell whether her tea was poured into the cup before the milk or after. The idea of the test was that distribution told him the best possible guesses that could be made for the experiment he designed.

Doctor Bristol-Roach was presented with eight cups of tea to taste, some poured milk first and some not. The lady then tasted them one-by-one reporting which type she believed each cup was. History records that the well bred biologist was more than validated by the experiment.