General Multiplication Rule

Suppose you have three cards (labeled A, B, and C) face down and you randomly select one of them. Assume that the one you selected was labeled A. Now randomly select one of the remaining cards. What is the probability that it is labeled B given that you are holding the one labeled A? Since there are two remaining cards, the probability is 1/2.

We say that P(B given that A has occurred) = 1/2

The notation for this is

The vertical line in this notation stands for "given"

General Multiplication Rule

Assume that A and B are events. Then

Example 1 (without replacement)

A box contains 5 marbles which are identical in size. Two of the marbles are green and three are red. Select two marbles without replacement. Find the probability that you get a red marble followed by a green one.

Solution .

Let R stand for getting a red marble on the first draw and G stand for getting a green marble on the second draw. We must find P(R and G). Note that P(R)= 3/5. Once the first marble is selected there are 4 left in the box. Since we are assuming that the first one drawn was red, two of the remaining 4 are red. Thus,

Hence we have

Example 2 (with replacement)

A box contains 5 marbles which are identical in size. Two of the marbles are green and three are red. Select a marble, record its color, and replace it. Do this a second time. Find the probability that you get a red marble followed by a green one.

Solution .

Let R stand for getting a red marble on the first draw and G stand for getting a green marble on the second draw. We must find P(R and G). Note that P(R)= 3/5. After the first marble is selected and replaced in the box, there are still 5 marbles in the box. Thus,

and

Note that in example two the probability of green given that the first was red is the same as drawing a green marble, 3/5. In this case the fact that we drew a red marble on the first draw did not affect the probability of drawing a green one on the second draw. The events in example 2 are said to be independent.

Contrast this with example one. The probability of getting a green on the second given that we had a red on the first draw was 2/4 which is not the same probability as drawing a green. The events in example one are dependent .

Definition

Two events A and B are said to be independent if the occurence of A does not affect the probability of B.

Another way to look at this is that A and B are independent if

Example 3

Two cards are randomly selected without replacement from a deck of cards. Find the probability that they are both aces.

Solution .

Notice that the numerator of the second fraction is three since we've assumed that we got an ace for the first card and that the denominator is 51 since we've already selected one card.