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Copyright © Cengage Learning. All rights reserved. 7.3 Rules of Probability Copyright © Cengage Learning. All rights reserved.

Properties of the Probability Function and Their Applications

Properties of the Probability Function and Their Applications Let S be a sample space of an experiment and suppose E and F are events of the experiment.

Properties of the Probability Function and Their Applications We have the following properties: If E and F are mutually exclusive events, then P(E  F) = P(E) + P(F) Figure 7

Properties of the Probability Function and Their Applications Property 3 may be easily extended to the case involving any finite number of mutually exclusive events. Thus, if E1, E2, . . . , En are mutually exclusive events, then P(E1  E2  ···  En) = P(E1) + P(E2) + ··· +P(En)

Applied Example 1 – SAT Verbal Scores The superintendent of a metropolitan school district has estimated the probabilities associated with the SAT verbal scores of students from that district. The results are shown in Table 7. Table 7

Applied Example 1 – SAT Verbal Scores cont’d If a student is selected at random, what is the probability that his or her SAT verbal score will be: a. More than 400? b. Less than or equal to 500? c. Greater than 400 but less than or equal to 600?

Applied Example 1 – Solution Let A, B, C, D, E, and F denote, respectively, the event that the score is greater than 700, greater than 600 but less than or equal to 700, greater than 500 but less than or equal to 600, and so forth. Then these events are mutually exclusive. Therefore, a. The probability that the student’s score will be more than 400 is given by P(D  C  B  A) = P(D) + P(C) + P(B) + P(A)

Applied Example 1 – Solution cont’d = .23 + .19 + .07 + .01 = .5 b. The probability that the student’s score will be less than or equal to 500 is given by P(D  E  F) = P(D) + P(E) + P(F) = .23 + .31 + .19 = .73

Applied Example 1 – Solution cont’d c. The probability that the student’s score will be greater than 400 but less than or equal to 600 is given by P(C  D) = P(C) + P(D) = .19 + .23 = .42

Properties of the Probability Function and Their Applications Property 3 holds if and only if E and F are mutually exclusive. In the general case, we have the following rule:

Example 2 A card is drawn from a well-shuffled deck of 52 playing cards. What is the probability that it is an ace or a spade?

Example 2 – Solution  

Applied Example 3 – Quality Control The quality-control department of Vista Vision, manufacturer of the Pulsar 42-inch plasma TV, has determined from records obtained from the company’s service centers that 3% of the sets sold experience video problems, 1% experience audio problems, and 0.1% experience both video and audio problems before the expiration of the 90-day warranty. Find the probability that a plasma TV purchased by a consumer will experience video or audio problems before the warranty expires.

Applied Example 3 – Solution Let E denote the event that a plasma TV purchased will experience video problems within 90 days, and let F denote the event that a plasma TV purchased will experience audio problems within 90 days. Then P(E) = .03 P(F) = .01 P(E  F) = .001

Applied Example 3 – Solution cont’d The event that a plasma TV purchased will experience video problems or audio problems before the warranty expires is E  F, and the probability of this event is given by 0.39 (refer Figure 10). P(E  F) = P(E) + P(F) – P(E  F) = .03 + .01 – .001 = .039 P(E  F) = P(E) + P(F) – P(E  F) Figure 10

Properties of the Probability Function and Their Applications Here is another property of a probability function that is of considerable aid in computing the probability of an event: Property 5 is an immediate consequence of Properties 2 and 3.

Properties of the Probability Function and Their Applications Indeed, we have E  Ec = S and E  Ec = Ø, so 1 = P(S) = P(E  Ec) = P(E) + P(Ec) and therefore, P(Ec) = 1 – P(E)

Applied Example 4 – Warranties What is the probability that a Pulsar 42-inch plasma TV bought by a consumer will not experience video or audio difficulties before the warranty expires? Solution: Let E denote the event that a plasma TV bought by a consumer will experience video or audio difficulties before the warranty expires.

Applied Example 4 – Solution cont’d Then the event that the plasma TV will not experience either problem before the warranty expires is given by Ec, with probability P(Ec) = 1 – P(E) = 1 – .039 = .961

Practice p. 386 Self-Check Exercises #2