Events and Their Probabilities

In the introduction to this chapter we used the term event much as it would be used in everyday language. Then, in Section 4.1 we introduced the concept of an experiment and its associated experimental outcomes or sample points. Sample points and events provide the foundation for the study of probability. As a result, we must now introduce the formal definition of an event as it relates to sample points. Doing so will provide the basis for determining the probability of an event.

EVENT

An event is a collection of sample points.

For an example, let us return to the KP&L project and assume that the project manager is interested in the event that the entire project can be completed in 10 months or less. Referring to Table 4.3, we see that six sample points—(2, 6), (2, 7), (2, 8), (3, 6), (3, 7), and (4, 6)—provide a project completion time of 10 months or less. Let C denote the event that the project is completed in 10 months or less; we write

C = {(2, 6), (2, 7), (2, 8), (3, 6), (3, 7), (4, 6)}

Event C is said to occur if any one of these six sample points appears as the experimental outcome.

Other events that might be of interest to KP&L management include the following.

L = The event that the project is completed in less than 10 months

M = The event that the project is completed in more than 10 months

Using the information in Table 4.3, we see that these events consist of the following sample points.

L = {(2, 6), (2, 7), (3, 6)}

M = {(3, 8), (4, 7), (4, 8)}

A variety of additional events can be defined for the KP&L project, but in each case the event must be identified as a collection of sample points for the experiment.

Given the probabilities of the sample points shown in Table 4.3, we can use the follow­ing definition to compute the probability of any event that KP&L management might want to consider.

PROBABILITY OF AN EVENT

The probability of any event is equal to the sum of the probabilities of the sample points in the event.

Using this definition, we calculate the probability of a particular event by adding the probabilities of the sample points (experimental outcomes) that make up the event. We can now compute the probability that the project will take 10 months or less to complete. Be­cause this event is given by C = {(2, 6), (2, 7), (2, 8), (3, 6), (3, 7), (4, 6)}, the probability of event C, denoted P(C), is given by

P(C) = P(2, 6) + P(2, 7) + P(2, 8) + P(3, 6) + P(3, 7) + P(4, 6)

Refer to the sample point probabilities in Table 4.3; we have

P(C) = .15 + .15 + .05 + .10 + .20 + .05 = .70

Similarly, because the event that the project is completed in less than 10 months is given by L = {(2, 6), (2, 7), (3, 6)}, the probability of this event is given by

P(L) = P(2, 6) + P(2, 7) + P(3, 6)

= .15 + .15 + .10 = .40

Finally, for the event that the project is completed in more than 10 months, we have M = {(3, 8), (4, 7), (4, 8)} and thus

P(M) = P(3, 8) + P(4, 7) + P(4, 8)

= .05 + .10 + .15 = .30

Using these probability results, we can now tell KP&L management that there is a .70 probability that the project will be completed in 10 months or less, a .40 probability that the project will be completed in less than 10 months, and a .30 probability that the project will be completed in more than 10 months. This procedure of computing event probabilit­ies can be repeated for any event of interest to the KP&L management.

Any time that we can identify all the sample points of an experiment and assign proba­bilities to each, we can compute the probability of an event using the definition. However, in many experiments the large number of sample points makes the identification of the sample points, as well as the determination of their associated probabilities, extremely cumbersome, if not impossible. In the remaining sections of this chapter, we present some basic probability relationships that can be used to compute the probability of an event without knowledge of all the sample point probabilities.

Source:  Anderson David R., Sweeney Dennis J., Williams Thomas A. (2019), Statistics for Business & Economics, Cengage Learning; 14th edition.