Probability question

Question

A shipment of 8 similar microcomputers to a retail outlet contains 3 that are defective. If a school makes a random purchase of 2 of these computers, find the probability mass function for the number of defectives.

Answer 1

This can be built up from first principles and combinations. Its official name is a hypergeometric random variable. Consider the basic probability ratio: the number of successes divided by the number of possible outcomes. We'll consider both, one at a time.

First let's consider the denominator. We are selecting 2 computers from a total of 8. This is ₈C₂, which is the total number of possible outcomes.

Next, let's look at the numerator, which is the number of successes. We do not have a specified number for how many are defective, so this is our variable x. How many defective computers are there to choose from? Since there are 3, there are ₃C_x possible selections of x computers out of 3 defective ones.

But this does not tell the whole story. Just as there are 3 defective computers, there are 5 non-defective ones. Whatever remains of the 2 computers selected must come from the 5 non-defective computers. In combination language, this is ₅C_2-x.

So here's the final product:

(₃C_x) (₅C_2-x)

P(X = x) = ---------------------

(₈C₂)