The complement of an event A, denoted by , is the set of all outcomes in

Example 2.10 DEFINITION Sometimes A and B have no outcomes in common, so that the intersection of A and B contains no outcomes. Let denote the null event the event consisting of no outcomes whatsoever. When , A and B are said to be mutually exclusive or disjoint events. A ¨ B 5 [ [ A small city has three automobile dealerships: a GM dealer selling Chevrolets and Buicks; a Ford dealer selling Fords and Lincolns; and a Toyota dealer. If an experi- ment consists of observing the brand of the next car sold, then the events and are mutually exclusive because the next car sold cannot be both a GM product and a Ford product at least until the two companies merge. ■ The operations of union and intersection can be extended to more than two events. For any three events A, B, and C, the event is the set of outcomes contained in at least one of the three events, whereas is the set of out- comes contained in all three events. Given events , these events are said to be mutually exclusive or pairwise disjoint if no two events have any out- comes in common. A pictorial representation of events and manipulations with events is obtained by using Venn diagrams. To construct a Venn diagram, draw a rectangle whose interior will represent the sample space . Then any event A is represented as the interior of a closed curve often a circle contained in . Figure 2.1 shows examples of Venn diagrams. S S A 1 , A 2 , A 3 , c A ¨ B ¨ C A ´ B ´ C B 5 5Ford, Lincoln6 A 5 5Chevrolet, Buick6 EXERCISES Section 2.1 1–10 1. Four universities—1, 2, 3, and 4—are participating in a holi- day basketball tournament. In the first round, 1 will play 2 and 3 will play 4. Then the two winners will play for the championship, and the two losers will also play. One possi- ble outcome can be denoted by 1324 1 beats 2 and 3 beats 4 in first-round games, and then 1 beats 3 and 2 beats 4. a. List all outcomes in . b. Let A denote the event that 1 wins the tournament. List outcomes in A. c. Let B denote the event that 2 gets into the championship game. List outcomes in B. d. What are the outcomes in and in ? What are the outcomes in A⬘? 2. Suppose that vehicles taking a particular freeway exit can turn right R, turn left L, or go straight S. Consider A ¨ B A ´ B S observing the direction for each of three successive vehicles. a. List all outcomes in the event A that all three vehicles go in the same direction. b. List all outcomes in the event B that all three vehicles take different directions. c. List all outcomes in the event C that exactly two of the three vehicles turn right. d. List all outcomes in the event D that exactly two vehicles go in the same direction. e. List outcomes in D⬘, , and . 3. Three components are connected to form a system as shown in the accompanying diagram. Because the components in the 2–3 subsystem are connected in parallel, that subsystem will function if at least one of the two individual components C ¨ D C ´ D A B a Venn diagram of events A and B A B e Mutually exclusive events A B c Shaded region is A 傼 B A d Shaded region is A A B b Shaded region is A 傽 B Figure 2.1 Venn diagrams Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.