$$

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

$$

$$

\begin{array}{l}{\text { In most gambling games, the odds given are not fair. }} \\ {\text { For example, if the odds of winning are really } 20 \text { to } 1,} \\ {\text { the house might offer } 15 \text { to } 1 \text { in order to make a profit. }} \\ {\text { Odds can be expressed as a fraction or as a ratio, }} \\ {\text { such as } \frac{5}{1}, 5: 1, \text { or } 5 \text { to } 1 . \text { Odds are computed in favor }} \\ {\text { of the event or against the event. The formulas for }} \\ {\text { odds are }}\end{array}

$$

$$

\begin{array}{l}{\text { Odds in favor }=\frac{P(E)}{1-P(E)}} \\ {\text { Odds against }=\frac{P(\bar{E})}{1-P(\bar{E})}}\end{array}

$$

In the die example,

$$

\begin{array}{c}{\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}\end{array}

$$

Find the odds in favor of and against each event.

a. Rolling a die and getting a 2

b. Rolling a die and getting an even number

c. Drawing a card from a deck and getting a spade

d. Drawing a card and getting a red card

e. Drawing a card and getting a queen

f. Tossing two coins and getting two tails

g. Tossing two coins and getting exactly one tail