[superprismatic]

Consider a horse race at a racetrack. Further consider what

happens when people bet that a horse will win that race.

A percentage, X (<100%), of the total money bet on these horses

to win is returned to the people who have bet on the winning

horse. A person's share of the winnings is proportional to

how much he bet. The rest, (1-X) of the money bet, is used

to pay for expenses and prizes to the horse owners. This

is, basically, the definition of parimutuel betting.

Now, the racetrack continuously updates odds for each horse

on a large display for all to see. These updates stop at the

start of the race when no further betting is allowed. These

odds tell the bettors what the return on their bet would be

should their horse win that race. The odds are posted as a

fraction A/B which means that for each dollar the bettor bet,

he will receive $((A/B)+1) should his horse win. So, for

example, if the bettor bet $1 and the odds posted as 15/1 just

as the race began, the track would pay the bettor $16 ($15

plus a return of his $1 bet). Odds of 8/5 would return $2.60

($1.60 plus his $1 bet).

Suppose the odds posted at the start of a 7 horse race were

15/1, 7/2, 8/5, 12/1, 5/2, 6/1, and 10/1. You can derive a

reasonable estimate for X (defined in the first paragraph)

from this information. What value would you give X and why?

[Guest]

if i understand

x=.7900509183

[Guest]

if i understand

x=.7900509183

I agree.

First you convert each of the odds into a percentage by dividing the right (under) by the sum of the two numbers (over/Under).

After adding all of the odds up we get 126.6%.

The 26.6% is the houses keep, so we divide 100 by 126 and find that the betters receive about 79% of the pot back and the house keeps about 21%.

[Guest]

Here's another way to solve it...

Consider this:

So, total bet amount is Y

Y = a+b+c+d+e+f+g

Then the payout for each horse winning = Y (1 - X) = 16a = 4.5b = 2.6c = 13d = 3.5e = 7f = 11g

Take each of b to g in terms of a and you get

Y = 20.25 a

So,

20.25 a (1 - X) = 16 a

This gives X as 0.79005

And that is how the house retains about 21%

Edited July 23, 2009 by DeeGee

[Guest]

I think it's the other way arround... 1-X is 79%. So the X is 21%... isn't it? There is also a problem with assumption that the payout for each horse winning is the same.

[Guest]

I think it's the other way arround... 1-X is 79%. So the X is 21%... isn't it? There is also a problem with assumption that the payout for each horse winning is the same.

You are right. In the soln I posted, X is 21%. However, X was the money retained by the house and not the money paid to the winners (opposite from what was in the question).

As for the payout being the same, it is correct. No matter which horse wins the total payout is the same. The house keeps 21% of the bets made and distributes 79% of the bets. In fact, the house calculates the odds for horses after deducting its retention and looking at what would be winning per $ for each horse.

In case, I didnt explain well, you can see this link to get a better understanding:

http://en.wikipedia.org/wiki/Parimutuel_betting

Edited July 23, 2009 by DeeGee

[Guest]

No comment here...(insert sweat drop)