4-1 Basic Concepts of Probability 153 In the three definitions that follow, the actual odds against and the actual odds in favor reflect the actual likelihood of an event, but the payoff odds describe the payoff amounts that are determined by casino, lottery, and racetrack operators. Racetracks and casinos are in business to make a profit, so the payoff odds will not be the same as the actual odds. DEFINITIONS The actual odds against event A occurring are the ratio P1A2>P1A2, usually expressed in the form of a:b (or “a to b”), where a and b are integers. (Reduce using the largest common factor; if a = 16 and b = 4, express the odds as 4:1 instead of 16:4.) The actual odds in favor of event A occurring are the ratio P1A2>P1A2, which is the reciprocal of the actual odds against that event. If the odds against an event are a:b, then the odds in favor are b:a. The payoff odds against event A occurring are the ratio of net profit (if you win) to the amount bet: Payoff odds against event A = 1net profit2:1amount bet2 SOLUTION a. With P1132 = 1>38 and P1not 132 = 37>38, we get Actual odds against 13 = P1not 132 P1132 = 37>38 1>38 = 37 1 or 37:1 b. Because the casino payoff odds against 13 are 35:1, we have 35:1 = 1net profit2:1amount bet2 So there is a $35 profit for each $1 bet. For a $5 bet, the net profit is $175 (which is 5 * $35). The winning bettor would collect $175 plus the original $5 bet. After winning, the total amount collected would be $180, for a net profit of $175. c. If the casino were not operating for profit, the payoff odds would be changed to 37:1, which are the actual odds against the outcome of 13. With payoff odds of 37:1, there is a net profit of $37 for each $1 bet. For a $5 bet, the net profit would be $185. (The casino makes its profit by providing a profit of only $175 instead of the $185 that would be paid with a roulette game that is fair instead of favoring the casino.) Actual Odds Versus Payoff Odds EXAMPLE 9 If you bet $5 on the number 13 in roulette, your probability of winning is 1>38, but the payoff odds are given by the casino as 35:1. a. Find the actual odds against the outcome of 13. b. How much net profit would you make if you win by betting $5 on 13? c. If the casino was not operating for profit and the payoff odds were changed to match the actual odds against 13, how much would you win if the outcome were 13? YOUR TURN. Do Exercise 41 “Florida Pick 3.”
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