Suppose a bookmaker tells you the odds against the...

Suppose a bookmaker tells you the odds against the Washington Redskins beating the Dallas Cowboys next week is $3: 2$. Assuming the odds are correct, (a) what is the probability that the Redskins will win and (b) if the Redskins win and you have bet $$\$ 10$$ that they will win, how much will you win?

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Key Concepts

Odds Against

Odds represent the ratio of unfavorable outcomes to favorable outcomes in an event. When odds against an event are expressed as a ratio a:b, it means that for every b successful outcomes, there are a failures. This concept is foundational in probability and gambling as it provides a way to express the likelihood of an event occurring.

Conversion of Odds to Probability

To convert odds into probability, you consider the total number of possible outcomes, which is the sum of both favorable and unfavorable outcomes. For odds given as a:b against an event, the probability that the event will happen is calculated as b divided by (a + b). This conversion is essential for quantifying how likely an event is to occur based on the odds provided.

Betting Payout Calculation

Betting payout is calculated based on the amount wagered and the specified odds. In scenarios where odds are given as a ratio, such as a:b, a winning bet typically yields a profit proportionate to that ratio. This means that if you bet a certain amount, your profit will be determined by multiplying your bet by the ratio of unfavorable outcomes to favorable outcomes, ensuring that your return reflects the risk assumed as indicated by the odds.

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