Odds

When enquiring about the likelihood of an event we might ask, 'What are the odds?'

The odds is a comparison between the chances of an event occurring and not occurring. The comparison is expressed as a ratio like $4:1$4:1. In the case of a horse race, odds of $4:1$4:1 expresses the belief that a particular horse is $4$4 times as likely not to win the race as to win it.

In other situations, the odds may be more than a belief or guess about an outcome. Instead, the chances of an event occurring or not might be known precisely. In the case of a dice throwing experiment, for example, it is clear that there is one chance of getting a six on the throw of a dice and five chances of not getting a six. We would say the odds of getting a six are $1:5$1:5 or the odds of not getting a six are $5:1$5:1.

There is a connection between the odds convention and the more formal way of expressing probabilities as percentages or fractions. 

In the case of the dice experiment, we would say there are six possible outcomes in total and therefore the probability of throwing a six is $\frac{1}{6}$16​ and the probability of not throwing a six is $\frac{5}{6}$56​. The ratio of these two probabilities is the odds ratio, $1:5$1:5.

In the horse racing example, the chosen horse has been assigned a probability of four chances in five of not winning and one chance in five of winning . That is, the probabilities are $0.8$0.8 and $0.2$0.2 respectively. Again, the ratio of these probabilities is the odds ratio, $4:1$4:1,

 

Example 1

The weather report predicts rain with a probability of $30%$30%. What are the odds that it will rain?

The probability of no rain must be $70%$70%. So, the odds for rain are $30:70$30:70 or $3:7$3:7.

 

Example 2

In a certain board game, a player's token can land on various differently coloured regions of equal area. Five of the regions are coloured green, four are red, four are blue and two are black. If we assume that each region is equally likely, what are the odds of avoiding a black region? 

We see that there are two black regions and thirteen non-black regions. Therefore the odds of avoiding a black region are $13:2$13:2.

Expressed as probabilities, we would say the probability of landing on black is $\frac{2}{15}$215​ and the probability of landing on some other colour is $\frac{13}{15}$1315​.

 

In games of chance, the odds ratio indicates what the payoff should be. A $\$1$$1 bet on a horse with odds $100:1$100:1 wins $\$100$$100 if the horse wins and loses $\$1$$1 if the horse does not win.

 

More Worked Examples

Question 1

Question 2

Question 3