Odds Odds are used in gambling games to make them fair.

QUESTION:

Odds Odds are used in gambling games to make them fair. For example, if you rolled a die and won every time you rolled a 6, then you would win on average once every 6 times. So that the game is fair, the odds of 5 to 1 are given. This means that if you bet $1 and won, you could win $5. On average, you would win $5 once in 6 rolls and lose $1 on the other 5 rolls—hence the term fair game.

   In most gambling games, the odds given are not fair. For example, if the odds of winning are really 20 to 1, the house might offer 15 to 1 in order to make a profit.

Odds can be expressed as a fraction or as a ratio, such as \(\frac{5}{1}\), 5:1, or 5 to 1. Odds are computed in favor of the event or against the event. The formulas for odds are

\(\text { Odds in favor }=\frac{P(E)}{1-P(E)}\)

\(\text { Odds against }=\frac{P(\bar{E})}{1-P(\bar{E})}\)

In the die example,

\(\text { Odds in favor of a } 6=\frac{\frac{1}{6}}{\frac{5}{6}}=\frac{1}{5} \text { or } 1: 5\)

\(\text { Odds against a } 6=\frac{\frac{5}{6}}{\frac{1}{6}}=\frac{5}{1} \text { or } 5: 1\)