160 CHAPTER 4 Probability Whenever A and B are disjoint, P1A and B2 becomes zero in the formal addition rule, so for disjoint events A and B we have P1A or B2 = P1A2 + P1B2. But again, instead of blind use of a formula, it is better to understand and use the intuitive addition rule. Here is a summary of the key points of the addition rule: 1. To find P1A or B2, first associate the word or with addition. 2. To find the value of P1A or B2, add the number of ways A can occur and the number of ways B can occur, but be careful to add without double counting. Complementary Events and the Addition Rule In Section 4-1 we used A to indicate that event A does not occur. Common sense dictates this principle: We are certain (with probability 1) that either an event A occurs or it does not occur, so it follows that P1A or A2 = 1. Because events A and A must be disjoint, we can use the addition rule to express this principle as follows: P1A or A2 = P1A2 + P1A2 = 1 This result of the addition rule leads to the following three expressions that are “equivalent” in the sense that they are just different forms of the same principle. Disjoint Events EXAMPLE 2 Disjoint events: Event A—Randomly selecting someone for a clinical trial who is a male Event B—Randomly selecting someone for a clinical trial who is a female (The selected person cannot be both.) Events that are not disjoint: Event A—Randomly selecting someone taking a statistics course Event B—Randomly selecting someone who is a female (The selected person can be both.) YOUR TURN. Do Exercise 12 “Texting or Not Drinking.” RULE OF COMPLEMENTARY EVENTS P1A2 + P1A2 = 1 P1A2 = 1 - P1A2 P1A2 = 1 - P1A2 Smartphone Home EXAMPLE 3 Based on survey results from the Consumer Technology Association, the probability of randomly selecting a household in the United States and getting one with a smartphone is 0.87, so P1smartphone2 = 0.87. If a household is randomly selected, find the probability of getting one that does not have a smartphone. Go Figure 2300%: Increase in risk of crash when texting and driving. 600%: Increase in risk of crash when driving drunk.
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