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The odds for an event to occur is the ratio of the number of favourable outcomes to the number of unfavourable outcomes.
||\text{Odds For}=\dfrac{\text{Number of Favourable Outcomes}}{\text{Number of Unfavourable Outcomes}}|| -
The odds against an event occurring is the ratio of the number of unfavourable outcomes to the number of favourable outcomes.
||\text{Odds Against}=\dfrac{\text{Number of Unfavourable Outcomes}}{\text{Number of Favourable Outcomes}}||
Note: These definitions and formulas are valid when all possible outcomes are equiprobable.
We roll a |6|-sided die and are interested in the event "getting a |2| or a |5|".
a) What are the odds for?
b) What are the odds against?
a) There are |2| favourable outcomes: "getting a |2|" and "getting a |5|". The number of unfavourable outcomes is the number of possibilities of rolling a number other than |2| or |5.| Therefore, there are |4| unfavourable results. The odds for are |2:4| (or |1:2| as a reduced ratio).
b) Since there are |4| unfavourable outcomes and |2| favourable outcomes, the odds against are |4:2| (or |2:1| if we reduce the ratio).
The odds for and odds against are not probabilities. In fact, the value of the odds written as a fraction can be greater than |1,| which is never the case when calculating probability. However, it is possible to find the probability associated with these odds by dividing the favourable or unfavourable outcomes by the number of possible outcomes.
The notation used to represent the odds for and odds against can be confusing. In fact, the ratios can be written in the form |\dfrac{a}{b}| or in the form |a:b.| In both cases, it is a comparison between two quantities of the same nature.
A probability is a part of a whole. In this case, the notation used is generally |\dfrac{a}{b}.|
To avoid confusion on this page, we will use the notation |a:b| for the odds for and against and the notation |\dfrac{a}{b}| for probability.
If we have a ratio of odds for |a:b,| where |a| is the number of favourable outcomes and |b| is the number of unfavourable outcomes, then the probability is |\dfrac{a}{a+b}.|
A renowned sports analyst rates the home team's chances of winning its next game at |1:4.| A reporter picks up the story and claims that the probability of a win has been evaluated by the sports analyst to be |25\ \%.| Did the reporter report the information correctly?
The sports analyst believes that there is only one scenario where he sees the home team winning versus |4| possible scenarios where he sees them losing. The total number of possible scenarios in his prediction is therefore |1+4=5.|
The probability of a win is the ratio of the number of favourable outcomes to the number of possible outcomes.
||P(\text{Win})=\dfrac{1}{5}=20\ \%||
Answer: The journalist therefore did not correctly communicate the remarks of the sports analyst.
As we can see in the last example, odds of |1:4| are not the same as a probability of |\dfrac{1}{4}.| The following table shows a list of possible cases.
|
Odds for |
Probability (for) |
Odds against |
Probability (against) |
|---|---|---|---|
|
|1:1| |
|\dfrac{1}{2}=50\ \%| |
|1:1| |
|\dfrac{1}{2}=50\%| |
|
|1:2| |
|\dfrac{1}{3}=33.\overline{3}\ \%| |
|2:1| |
|\dfrac{2}{3}=66.\overline{6}\ \%| |
|
|1:3| |
|\dfrac{1}{4}=25\ \%| |
|3:1| |
|\dfrac{3}{4}=75\ \%| |
|
|1:4| |
|\dfrac{1}{5}=20\ \%| |
|4:1| |
|\dfrac{4}{5}=80\ \%| |
|
|1:5| |
|\dfrac{1}{6}=16.\overline{6}\ \%| |
|5:1| |
|\dfrac{5}{6}=83.\overline{3}\ \%| |
|
|1:9| |
|\dfrac{1}{10}=10\ \%| |
|9:1| |
|\dfrac{9}{10}=90\ \%| |
It is impossible to express a certain (sure) event in odds for, since the probability is equal to |1.|
In fact, if the probability is |1,| it means that there are no unfavourable outcomes. For example, if there are |3| favourable outcomes and |0| unfavourable outcomes, by applying the definition of the odds for, we get the ratio |3:0,| which is impossible, because we cannot divide by |0.|