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
$$
\begin{array}{l}
\text { Odds in favor }=\frac{P(E)}{1-P(E)} \\
\text { Odds against }=\frac{P(\bar{E})}{1-P(\bar{E})}
\end{array}
$$
In the die example,
$$
\begin{aligned}
&\text { Odds in favor of a } 6=\frac{\frac{1}{6}}{\frac{5}{6}}=\frac{1}{5} \text { or } 1: 5\\
&\text { Odds against a } 6=\frac{\frac{5}{6}}{\frac{1}{6}}=\frac{5}{1} \text { or } 5:
\end{aligned}
$$
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