11.2 Odds 675 The odds in favor of an event can also be determined without calculating probabilities by using the following formula. Odds in Favor of an Event In an experiment with equally likely outcomes, the odds in favor of an event E can also be determined using the following formula. = E E E Odds in favor of event number of outcomes favorable to number of outcomes unfavorable to In Example 3(a), the number of outcomes favorable to selecting a queen is 4 (the four queens), and the number of outcomes unfavorable to selecting a queen is 48 (all the other cards in the deck). Thus, Odds in favor of selecting a queen number of outcomes favorable to selecting a queen number of outcomes unfavorable to selecting a queen 4 48 1 12 = = = Notice both procedures give us the same odds in favor of selecting a queen, 1 to 12. Determining Probabilities from Odds When odds are given, either in favor of or against a particular event, it is possible to determine the probability that the event occurs and the probability that the event does not occur. The denominators of the probabilities are determined by adding the numbers in the odds statement. The numerators of the probabilities are the numbers given in the odds statements. Example 4 Determining Probabilities from Odds The odds against Ting being admitted to the college of her choice are 9 : 2. Determine the probability that (a) Ting is admitted and (b) Ting is not admitted. Solution a) We have been given odds against and have been asked to determine probabilities. = P P Odds against being admitted (fails to be admitted) (is admitted) Since the odds statement is 9 : 2, the denominators of both the probability of success and the probability of failure must be + 9 2, or 11. To get the odds ratio of 9 : 2, the probabilities must be 9 11 and . 2 11 Since odds against is a ratio of failure to success, the 9 11 and 2 11 represent the probabilities of failure and success, respectively. Thus, the probability that Ting is admitted (success) is . 2 11 b) The probability that Ting is not admitted (failure) is . 9 11 ■ Now try Exercise 45 Odds and probability statements are sometimes stated incorrectly. For example, consider the statement, “The odds of being selected to represent the district are 1 in 5.” Odds are given using the word to, not in. Thus, there is a mistake in this statement. The correct statement might be, “The odds of being selected to represent the district are 1 to 5” or “The probability of being selected to represent the district is 1 in 5.” Without additional information, it is not possible to tell which statement is the correct interpretation. Instructor Resources for Section 11.2 in MyLab Math • Objective-Level Videos 11.2 • PowerPoint Lecture Slides 11.2 • MyLab Exercises and Assignments 11.2
RkJQdWJsaXNoZXIy NjM5ODQ=