4.1 Hypergeometric distribution

The simplest probability density function is the hypergeometric. This is the most basic one because it is created by combining our knowledge of probabilities from Venn diagrams, the addition and multiplication rules, and the combinatorial counting formula.

To find the number of ways to get 2 aces from the four in the deck we computed:

And if we did not care what else we had in our hand for the other three cards we would compute:

Putting this together, we can compute the probability of getting exactly two aces in a 5 card poker hand as:

This solution is really just the probability distribution known as the Hypergeometric. The generalized formula is:

where x = the number we are interested in coming from the group with A objects.

h(x) is the probability of x successes, in n attempts, when A successes (aces in this case) are in a population that contains N elements. The hypergeometric distribution is an example of a discrete probability distribution because there is no possibility of partial success, that is, there can be no poker hands with 2 1/2 aces. Said another way, a discrete random variable has to be a whole, or counting, number only. This probability distribution works in cases where the probability of a success changes with each draw. Another way of saying this is that the events are NOT independent. In using a deck of cards, we are sampling WITHOUT replacement. If we put each card back after it was drawn then the hypergeometric distribution would not work.

For the hypergeometric to work,

1. the population must be dividable into two and only two independent subsets (aces and non-aces in our example). The random variable X = the number of items from the group of interest.
2. the experiment must have changing probabilities of success with each experiment (the fact that cards are not replaced after the draw in our example makes this true in this case). Another way to say this is that you sample without replacement and therefore each pick is not independent.
3. the random variable must be discrete, rather than continuous.

Chapter review

The combinatorial formula can provide the number of unique subsets of size x that can be created from n unique objects to help us calculate probabilities. The combinatorial formula is $\left(\begin{array}{c}n\\ x\end{array}\right) ={}_n{C}_x=\frac{n!}{x!(n-x)!}$

A hypergeometric experiment is a statistical experiment with the following properties:

1. You take samples from two groups.
2. You are concerned with a group of interest, called the first group.
3. You sample without replacement from the combined groups.
4. Each pick is not independent, since sampling is without replacement.

The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest. $h\left(x\right)=\frac{\left(\begin{array}{c}A\\ x\end{array}\right)\left(\begin{array}{c}N-A\\ n-x\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}$ .

Formula review

$h\left(x\right)=\frac{\left(\begin{array}{c}A\\ x\end{array}\right)\left(\begin{array}{c}N-A\\ n-x\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}$

Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample.