Question

‘Odds’ in horserace betting are defined as follows: 3/1 (three-to-one against) means a horse is expected to win once for every three times it loses; 3/2 means two wins out of five races; 4/5 (five to four on) means five wins for every four defeats, etc.

(a) Translate the above odds into ‘probabilities’ of victory.

(b) In a three-horse race, the odds quoted are 2/1, 6/4, and 1/1. What makes the odds different from probabilities? Why are they different?

(c) Discuss how much the bookmaker would expect to win in the long run at such odds, assuming each horse is backed equally.

Answer 1

(a) a/b odds mean a defeats and b win

therefore probability of winning is b/(a+b) since it is winning a times in (a+b) races

so for,

3/1 odds P(win) = 1/4

3/2 odds P(win) = 2/5

4/5 odds P(win) = 5/9

(b)Probability always ranges between 0 and 1. The odds are the ratio of the probability that the event will occur and the probability that the event will not occur. If the probability of an event occurring is P, then the probability of the event not occurring is 1-P.

Probability is used to compare the no. of success and the total no. of trials.

While odds is used to compare no. of success and no. of losses.

(c) let us say x is the amount of money put on each horse

if horse 1 wins :

bookmakers profit = (money put in bets) - (money paid)

= 3x - 2x = x

{2x is the winning amount for x dollars on horse 1)

if horse 2 wins :

bookmakers profit = (money put in bets) - (money paid)

= 3x - 1.5x = 1.5x

if horse 3 wins :

bookmakers profit = (money put in bets) - (money paid)

= 3x - x = 2x

bookmakers profit = profit(if horse 1 wins)*P(horse 1 win) + profit(if horse 2 wins)*P(horse 2 win) + profit(if horse 3 wins)*P(horse 3 win)

bookmakkers profit (total money bet 3x) = x*1/3 + 1.5x*2/5 + 2x*1/2

= x/3 + 3x/5 + x

= 1.93x

profit ratio = 1.93x / 3x = 0.64

therefore if total bets is 1$ equally divided among three horses profit in the long run for bookmaker is 0.64 $.