GMAT Quantitative Review

3.0 Math Review

3.1 Arithmetic

11. Discrete Probability

Many of the ideas discussed in the preceding three topics are important to the study of discrete probability. Discrete probability is concerned with experiments that have a finite number of outcomes. Given such an experiment, an event is a particular set of outcomes. For example, rolling a number cube with faces numbered 1 to 6 (similar to a 6-sided die) is an experiment with 6 possible outcomes: 1, 2, 3, 4, 5, or 6. One event in this experiment is that the outcome is 4, denoted {4}; another event is that the outcome is an odd number: {1, 3, 5}.

The probability that an event E occurs, denoted by P (E), is a number between 0 and 1, inclusive. If E has no outcomes, then E is impossible and image; if E is the set of all possible outcomes of the experiment, then E is certain to occur and image. Otherwise, E is possible but uncertain, and image. If F is a subset of E, then image. In the example above, if the probability of each of the 6 outcomes is the same, then the probability of each outcome is image, and the outcomes are said to be equally likely. For experiments in which all the individual outcomes are equally likely, the probability of an event E is

image.

In the example, the probability that the outcome is an odd number is

image.

Given an experiment with events E and F, the following events are defined: “not E” is the set of outcomes that are not outcomes in E“E or F” is the set of outcomes in E or F or both, that is, image“E and F” is the set of outcomes in both E and F, that is, image.

The probability that E does not occur is image. The probability that “E or F” occurs is image, using the general addition rule at the end of section 3.1.9 (“Sets”). For the number cube, if E is the event that the outcome is an odd number, {1, 3, 5}, and F is the event that the outcome is a prime number, {2, 3, 5}, then image and so image.

Note that the event “E or F” is image, and hence image.

If the event “E and F” is impossible (that is, image has no outcomes), then E and F are said to be mutually exclusive events, and image. Then the general addition rule is reduced to image.

This is the special addition rule for the probability of two mutually exclusive events.

Two events A and B are said to be independent if the occurrence of either event does not alter the probability that the other event occurs. For one roll of the number cube, let image and let image. Then the probability that A occurs is image, while, presuming B occurs, the probability that A occurs is

image.

Similarly, the probability that B occurs is image, while, presuming A occurs, the probability that B occurs is

image.

Thus, the occurrence of either event does not affect the probability that the other event occurs. Therefore, A and B are independent.

The following multiplication rule holds for any independent events E and Fimage.

For the independent events A and B above, image.

Note that the event “A and B” is image, and hence image. It follows from the general addition rule and the multiplication rule above that if E and F are independent, then

image.

For a final example of some of these rules, consider an experiment with events A, B, and C for which imageimage, and image. Also, suppose that events A and B are mutually exclusive and events B and C are independent. Then

image

image

Note that P (A or C) and P (A and C) cannot be determined using the information given. But it can be determined that A and C are not mutually exclusive since image, which is greater than 1, and therefore cannot equal P (A or C); from this it follows that image. One can also deduce that image, since image is a subset of A, and that image since C is a subset of image. Thus, one can conclude that image and image.