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Year 10 Interactive Maths - Second Edition


Odds

In general:

Odds in favour of event A occurring = Pr(A) / Pr(A')          Odds against event A occurring = Pr(A') / Pr(A)

The odds against winning are found by calculating:

Probability of losing / Probability of winning

If the probability that a horse will win a race is estimated to be 1/10, then the horse is given 1 chance of winning and 9 chances of losing out of 10 chances.

Let A be the event of winning (i.e. A' is the event of losing).

Now, odds against event A = Probability of losing / Probability of winning = Pr(A') / Pr(A) = 9 / 1

The odds against winning (i.e. the odds the horse will lose) are 9:1. We usually say the 'odds on' to win are 9:1.

Bookmakers assign odds against winning events in sport, horse racing, greyhound racing etc. So, if a bookmaker assigns odds of 9:1 without allowing for a profit margin, then a competitor is given 9 chances of losing and one chance of winning out of 10 chances. That is:

Pr(losing) = 9/10 = 0.9,          Pr(winning) = 1/10 = 0.1

Note that if a gambler places a $1 bet with a bookmaker at 9:1 odds on to win and his horse wins, the gambler will win $9 plus the $1 bet, obtaining $10; otherwise the gambler will lose $1. If a $5 bet is placed with the bookmaker at 9:1 and his horse wins, the gambler will win $45 plus the $5 bet, obtaining $50; otherwise the gambler will lose $5.

Past performance, the recent form of competitors, the expected weather conditions for the event, the jockey and other relevant factors are taken into consideration to determine the odds before the occurrence of the event. As gamblers place their bets, bookmakers adjust the odds in order to minimise the amounts paid out and thus maximise their profit, i.e. the bookmakers alter the odds so that they make a profit. The bookmakers' odds are decided on subjective estimates of probabilities biased in their favour.


Example 8

What are the odds in favour of throwing a 1 with a die?

Solution:

S = {1, 2, 3, 4, 5, 6}.  Let A be the event of obtaining 1.  Pr(A) = 1/6, Pr(A') = 1 - Pr(A) = 1 - 1/6 = 5/6

So odds in favour of event A = Pr(A) / Pr(A') = 1/5.  So, the odds in favour of throwing a 1 are 1 : 5.


Example 9

If there is a 30% chance of winning, find the odds against winning.

Solution:

Pr(winning) = 30/100 = 3/10,     Pr(losing) = 1 - Pr(winning) = 1 - 3/10 = 7/10

Odds against winning = Pr(losing) / Pr(winning) = 7/3

So, the odds against winning are 7:3, i.e. 'odds on' to win are 7:3.


Example 10

If the odds against winning a horse race are 2:1, find the probability of winning the race.

Solution:

Odds of 2:1 against winning means 2 losing chances to every winning chance. So, if the race is held 3 times, one would be expected to lose 2 races and win 1 race. Thus, the probability of winning the race is 1/3.


Example 11

If the odds in favour of winning a race are 3:5, find the probability of winning the race.

Solution:

Odds in favour of winning a race of 3:5 mean 3 chances of winning to every 5 chances of losing. So, if the race were held 8 times, one would be expected to win 3 races and lose 5 races. Thus, the probability of winning the race is 3/8.


Example 12

I bet $4 on a horse race. How much would I receive from the bookmaker if a win results with the following odds on to win?

a.  evens
b.  3:1
c.  5:2

Solution:

(a)  If the odds are evens, i.e. 1:1, I would win $4; and receive $8.     (b)  If the odds are 3:1, I would win $12; and receive $16.     (c)  If the odds are 5:2, i.e. 5/2, I would win $10; and receive $14.


Example 13

If there is a 0% chance of winning, can we find the odds against winning?

Solution:

Pr(winning) = 0/100 = 0     Pr(losing) = 1 - Pr(winning) = 1 - 0 = 0     Odds against winning = Pr(losing) / Pr(winning) = 1/0 = undefined

So, we cannot find the odds against winning.


Key Terms

odds, odds against winning, odds on to win


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