11.2 Odds 671 Odds SECTION 11.2 LEARNING GOALS Upon completion of this section, you will be able to: ■ Understand odds against an event. ■ Understand odds in favor of an event. ■ Understand how to obtain probabilities from odds and odds from probabilities. Why This Is Important In addition to applications involving sports and gambling, odds play an important role in many areas of society. Companies use odds when making difficult business decisions. Odds play an important role throughout the financial services industry, especially in all types of insurance. Odds are used when studying the spread of disease and the effectiveness of medical treatments. Odds are also used by social researchers when analyzing the results of surveys. A basic understanding of odds is important for many areas of employment. Odds Against an Event In addition to probability, odds provide us with another way to measure the chance that an event occurs. The odds given at gambling venues are usually given as odds against winning unless they are otherwise specified. The odds against an event is a ratio of the probability that the event will fail to occur (failure) to the probability that the event will occur (success). The odds against winning the Mega Millions lottery are about 300 million to 1. The odds against being audited by the IRS this year are about 262 to 1. The odds against the Washington Capitals winning the Stanley Cup this year may be 55 to 1. We see the word odds daily in newspapers and magazines and often use it ourselves. Yet there is widespread misunderstanding of its meaning. In this section, we will explain the meaning of odds. We will also discuss how to determine odds against an event and how to determine odds in favor of an event. Recreational Mathematics 87. Cola Preference Can people distinguish Coke from Pepsi? Which do they prefer? a) Design an experiment to determine the empirical probability that a randomly selected person can select Coke when given samples of both Coke and Pepsi. Answers will vary. b) Perform the experiment in part (a) and determine the empirical probability that a person can select Coke. Answers will vary. c) Determine the empirical probability that a person randomly selected will prefer Coke over Pepsi. Answers will vary. 88. Dice On a die, the sum of the dots on the opposite faces is seven. Two six-sided dice are placed together on top of one another, on a table, as shown in the following figure. If you walk around the table, what is the sum of all the dots on all the visible faces of the dice? 29 dots Research Activities 89. Insurance Write a paper on how insurance companies use empirical probabilities in determining insurance premiums. An insurance agent may be able to direct you to a source of information. 90. Bernoulli Family On page 657, we briefly discuss Jacob Bernoulli. The Bernoulli family produced several prominent mathematicians, including Jacob I, Johann I, and Daniel. Write a paper on the Bernoulli family, indicating some of the accomplishments of each of the three Bernoullis named and their relationship to one another. Indicate which Bernoulli the Bernoulli numbers are named after, which Bernoulli the Bernoulli theorem in statistics is named after, and which Bernoulli the Bernoulli theorem of fluid dynamics is named after. 91. Hemophilia in Royalty The effect of genetic inheritance is dramatically demonstrated by the occurrence of hemophilia in some of the royal families of Europe. Do research and write a report on the genetic causes of hemophilia and how the disease has affected royals starting with Queen Victoria (1819–1901) of the United Kingdom. Include in the report how probability can be used to predict those who become carriers and those who will have the disease. JaiAgnish/Shutterstock
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