## 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 ; if *E* is the set of all possible outcomes of the experiment, then *E* is *certain* to occur and . Otherwise, *E* is possible but uncertain, and . If *F* is a subset of *E*, then . In the example above, if the probability of each of the 6 outcomes is the same, then the probability of each outcome is , 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

.

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

.

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, ; *“E and F*” is the set of outcomes in both *E* and *F*, that is, .

The probability that *E* does not occur is . The probability that “*E* or *F*” occurs is , 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 and so .

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

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

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 and let . Then the probability that *A* occurs is , while, *presuming B occurs*, the probability that *A* occurs is

.

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

.

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 *F*: .

For the independent events *A* and *B* above, .

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

.

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

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 , which is greater than 1, and therefore cannot equal *P* (*A* or *C*); from this it follows that . One can also deduce that , since is a subset of *A*, and that since *C* is a subset of . Thus, one can conclude that and .