Hypergeometric Distribution

Given random variable :

where is the number of successes in a sequence of trials without replacement from a finite population of size that contain exactly possible successes.

In an experiment of trials without replacement from a population of with possible successes, the probability of successes is given by:

(see choose)

We choose successes from a pool of successes AND we draw fails from a pool of fails. Then we total it by the total number of ways to arrange the objects.

You may note that this is just standard probability (see choose examples).

If is large and is small, then we have:

Since the number of total items is so big and we're sampling so little, whether we replace or not makes little difference.

(Binomial Distribution)

Suppose we have 1500 marbles in a bag, 600 red and 900 blue. We will randomly draw 8 of those marbles. What is the probability that we will draw 3 red marbles?

Without replacement
Let be the number of red marbles. Then:

With replacement
Let be the number of red marbles. Then:

Given :

Suppose we have 15 marbles in a bag, 6 red and 9 blue. We will randomly draw 8 of those marbles without replacement. What is the probability that we draw 3 red marbles? What about with replacement?

solution
Without Replacement
Let be the number of red marbles drawn. Then and

With Replacement
Let be the number of red marbles drawn. Then (Binomial Distribution) and