672 CHAPTER 11 Probability In Example 1, when a die is rolled there are 6 possible equally likely outcomes: 1, 2, 3, 4, 5, and 6. If all the equally likely outcomes of an experiment are known, the odds against an event can be determined without calculating probabilities by using the following formula. Odds Against an Event The following formula may be used to determine the odds against an event. = = P P P P Odds against event (event fails to occur) (event occurs) (failure) (success) Example 1 Rolling a 4 Determine the odds against rolling a 4 on one roll of a fair die. Solution Before we can determine the odds, we must first determine the probability of rolling a 4 (success) and the probability of not rolling a 4 (failure). When a die is rolled there are six possible outcomes: 1, 2, 3, 4, 5, and 6. = = P P (rolling a 4) 1 6 (failure to roll a 4) 5 6 Now that we know the probabilities of success and failure, we can determine the odds against rolling a 4. = = = ⋅ = P P Odds against rolling a 4 (failure to roll a 4) (rolling a 4) 5 6 1 6 5 6 6 1 5 1 1 1 The ratio 5 1 is commonly written as 5:1 and is read “5 to 1.” Thus, the odds against rolling a 4 are 5 to 1. ■ Now try Exercise 15 Timely Tip The denominators of the probabilities in an odds problem will always divide out, as was shown in Example 1. Odds Against an Event In an experiment with equally likely outcomes, the odds against an event E can also be determined using the following formula. = E E E Odds against event number of outcomes unfavorable to number of outcomes favorable to In Example 1, using probabilities we determined the odds against rolling a 4 when a die is rolled. We can also determine the odds against rolling a 4 using the above definition. There are 5 outcomes that are unfavorable to rolling a 4 (1, 2, 3, 5, and 6). There is 1 outcome favorable to rolling a 4 (the 4 itself). Thus, Odds against rolling a 4 number of outcomes unfavorable to rolling a 4 number of outcomes favorable to rolling a 4 5 1 = = Therefore, the odds against rolling a 4 are 5 to 1. If all of the equally likely outcomes of an experiment are known, this alternate procedure for determining the odds against an event is often easier than using probabilities.
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