4-3 Complements, Conditional Probability, and Bayes’ Theorem 171 32.Same Birthdays If 25 people are randomly selected, find the probability that no 2 of them have the same birthday. Ignore leap years. 33.Exclusive Or The exclusive or means either one or the other event occurs, but not both. a. For the formal addition rule, rewrite the formula for P1Aor B2 assuming that the addition rule uses the exclusive or instead of the inclusive or. b. Repeat Exercise 11 “Texting or Drinking” using the exclusive or instead of the inclusive or. 34.Complements and the Addition Rule Refer to the table used for Exercises 9–20. Assume that one driver is randomly selected. Let A represent the event of getting a driver who texted while driving and let B represent the event of getting a driver who drove when drinking alcohol. Find P1Aor B2, find P1Aor B2, and then compare the results. In general, does P1Aor B2 = P1Aor B2? PART 1 Complements: The Probability of “At Least One” When finding the probability of some event occurring “at least once,” we should understand the following: ■ “At least one” = “one or more.” ■ Complement of “at least one” particular event = no occurrences of that event. For example, not getting at least 1 girl in 10 births is the same as getting no girls, which is also the same as getting 10 boys. Not getting at least 1 girl in 10 births = Getting no girls = Getting 10 boys The following steps describe the details of finding the probability of getting at least one of some event: Finding the probability of getting at least one of some event: 1. Let A = getting at least one of some event. 2. Then A = getting none of the event being considered. 3. Find P1A2 = probability that event A does not occur. (This is relatively easy using the multiplication rule.) 4. Subtract the result from 1: P1at least one occurrence of event A2 = 1 − P1nooccurrences of event A2 Key Concept In Part 1 of this section we extend the use of the multiplication rule to include the probability that among several trials, we get at least one of some specified event. In Part 2 we consider conditional probability: the probability of an event occurring when we have additional information that some other event has already occurred. In Part 3 we provide a brief introduction to Bayes’ theorem. 4-3 Complements, Conditional Probability, and Bayes’ Theorem o d rd. Probability of an Event That Has Never Occurred Some events are possible but are so unlikely that they have never occurred. Here is one such problem of great interest to political scientists: Estimate the probability that your single vote will determine the winner in a U.S. presidential election. Andrew Gelman, Gary King, and John Boscardin write in the Journal of the American Statistical Association (Vol. 93, No. 441) that “the exact value of this probability is of only minor interest, but the number has important implications for understanding the optimal allocation of campaign resources, whether states and voter groups receive their fair share of attention from prospective presidents, and how formal ‘rational choice’ models of voter behavior might be able to explain why people vote at all.” The authors show how the probability value of 1 in 10 million is obtained for close elections.
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