Page 89 - Probability Demystified

78 CHAPTER 5 Odds and Expectation

Odds

Odds are used by casinos, racetracks, and other gambling establishments to
determine payoffs when bets are made. For example, at a race, the odds that
a horse wins the race may be 4 to 1. In this case, if you bet $1 and the horse
wins, you get $4. If you bet $2 and the horse wins, you get $8, and so on.
Odds are computed from probabilities. For example, suppose you roll a
die and if you roll a three, you win. If you roll any other number, you lose.
Furthermore, if you bet one dollar and win, what would the payoff be if you
win? In this case, there are six outcomes, and you have one chance (outcome)
1
of winning, so the probability that you win is . That means on average you
6
win once in every six rolls. So if you lose on the first five rolls and win on the
sixth, you have lost $5 and therefore, you should get $5 if you win on the
sixth roll. So if you bet $1 and win $5, the odds are 1 to 5. Of course, there is
no guarantee that you will win on the sixth roll. You may win on the first roll
or any roll, but on average for every six rolls, you will win one time over the
long run.
In gambling games, the odds are expressed backwards. For example,
if there is one chance in six that you will win, the odds are 1 to 5, but in
general, the odds would be given as 5 to 1. In gambling, the house (the people
running the game) will offer lower odds, say 4 to 1, in order to make a profit.
In this case, then, the player wins on average one time in every 6 rolls and
spends on average $5, but when the player wins, he gets only $4. So the house
wins on average $1 for every six rolls of the player.
1
Odds can be expressed as a fraction, , or as a ratio, 1 : 5. If the odds of
5
winning the game are 1 : 5, then the odds of losing are 5 : 1. The odds of
winning the game can also be called the odds ‘‘in favor’’ of the event
occurring. The odds of losing can also be called ‘‘the odds against’’ the event
occurring.
The formulas for odds are

PðEÞ
odds in favor ¼
1 PðEÞ
PðEÞ
odds against ¼
1 PðEÞ

where P(E) is the probability that the event E occurs and PðEÞ is the
probability that the event does not occur.