11.2 Odds 673 In Example 1, we considered the possible outcomes of the die: 1, 2, 3, 4, 5, 6. Over the long run, one of every six rolls will result in a 4, and five of every six rolls will result in a number other than 4. Therefore, if a person is gambling, for each dollar bet in favor of the rolling of a 4, $5 should be bet against the rolling of a 4 if the person is to break even. The person betting in favor of the rolling of a 4 will either lose $1 (if a number other than a 4 is rolled) or win $5 (if a 4 is rolled). The person betting against the rolling of a 4 will either win $1 (if a number other than a 4 is rolled) or lose $5 (if a 4 is rolled). If this game is played for a long enough period, each player theoretically will break even. Example 2 involves a circle graph that involves percents; see Fig. 11.4, below. Before we discuss Example 2, let us briefly discuss percents. Recall that probabilities are numbers between 0 and 1, inclusive. We can change a percent between 0% and 100% to a probability by writing the percent as a fraction or a decimal number as discussed in Section 10.1. In Fig. 11.4, we see that the green sector of the circle, Lee , corresponds to 13%. To change 13% to a probability, we can write , 13 100 or 0.13. Note that both the fraction and the decimal number are numbers between 0 and 1, inclusive. Example 2 Blue Jeans In a recent survey, students at Etheridge University were asked to name the brand of blue jeans they preferred. The circle graph in Fig. 11.4, which was created using StatCrunch , shows the results of the survey. If a student from the survey is randomly selected, determine the odds against the individual preferring Favorite Brand of Blue Jeans Carhartt, 4% Levi’s, 35% Diesel, 17% Lucky, 11% Lee, 13% Wrangler, 15% Other, 5% Figure 11.4 a) Levi’s. b) Wrangler Solution a) From the graph, we see that 35%, or = , 35 100 7 20 of the students surveyed prefer Levi’s blue jeans. Therefore, the probability of a student preferring Levi’s blue jeans is . 7 20 The probability of a student not preferring Levi’s blue jeans is − = 1 . 7 20 13 20 = = P P Odds against a student preferring Levi’s (student not preferring Levi’s) (student preferring Levi’s) 13 20 7 20 = ⋅ = 13 20 20 7 13 7 , or 13 : 7 Thus, the odds against a student preferring Levi’s blue jeans are 13 : 7. Did You Know? Gambling in Ancient Times A rchaeologists have found evidence of gambling in all cultures, from the Aboriginal People living in what is now Australia during the Stone Age, to the ancient Egyptians, and across cultures touched by the Roman Empire. There is an equally long history of moral and legal opposition to gambling.
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