02/20/2020
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Expressions of likelihood are often given as odds, such as \( 50:1 \) (or “50 to 1”).
Because the use of odds makes many calculations difficult, statisticians, mathematicians, and scientists prefer to use probabilities.
The advantage of odds is that they make it easier to deal with money transfers associated with gambling, so they tend to be used in casinos, lotteries, and racetracks.
Note – in the three definitions that follow, the actual odds against and the actual odds in favor are calculated with the actual likelihood of some event;
The actual odds correspond to actual probabilities of outcomes, but the payoff odds are set by racetrack and casino operators.
Actual odds against event \( A \) – this is the probability of event \( \overline{A} \) relative to the event \( A \), i.e.,
\[ \frac{P\left(\overline{A}\right)}{P(A)} \]
Actual odds in favor of event \( A \) – this is the probability of event \( A \) relative to the event \( \overline{A} \), i.e.,
\[ \frac{P(A)}{P\left(\overline{A}\right)} \]
Payoff odds against event \( A \) – this is the ratio of net profit (if you win) to the amount bet:
\[ \text{payoff odds against event }A = (\text{net profit}):(\text{amount bet}) \]
If you bet \( 5 \) dollars on the number \( 13 \) in roulette, your probability of winning is \( \frac{1}{38} \) and the payoff odds are given by the casino as \( 35:1 \).
Discuss with a neighbor: what are the actual odds for and the actual odds against winning with a bet on \( 13 \)?
Recall our formula for payoff odds, \[ \text{payoff odds against event }A = (\text{net profit}):(\text{amount bet}) \]
Discuss with a neighbor: how much net profit would you make if you win by betting on \( 13 \)?
If you bet \( 5 \) dollars on the number \( 13 \) in roulette, your probability of winning is \( \frac{1}{38} \) and the payoff odds are given by the casino as \( 35:1 \).
Discuss with a neighbor: 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 with a bet of \( 5 \) dollars if the outcome were \( 13 \)?
Courtesy of Bin im Garten CC via Wikimedia Commons
Courtesy of Bin im Garten CC via Wikimedia Commons