Gambling on Gladiators; a math question!

[thefirecrack3r]

I actually need help answering sort of a math question for future D&D games. My players recently competed in a gladiator style arena, now what I want to do is let them return to this place but not necessarily to compete. Gambling on champions and beasts is something that takes place in an organized fashion at this arena, On average about 1000 people are in attendance. I was thinking I could just look at how this works for horse races, but horse racing is also based on the horses win/loss ratio and other things. What I want instead is to come up with a system based purely on the amount of cash laid down on each contestant. So for instance, 3 champions are about to duke it out, 1,000 gp on contestant one (hence forth C1) 2,500 on C2 and a whopping 4,500 on C3. The player bet 50 gold on C2, who wins the fight, how do I determine what his pay out is? I'm not math savvy at all so I do need help here. I want him to essentially earn a % based on A)the amount of gold he put down and B)the amount of gold lost by those betting on C1 and C3. Make sense? I'll be able to do the math once I have an equation but right now I'm just not connecting the dots on how this would work.
*Of course in reality the arena owners are taking a small % of the losers bets but I'm not going to ask for that to be calculated for simplicity's sake.

[JusticeZero]

Sell tickets for a fixed cost. Pool all the bets. Organizers skim 10%. Winners take the remainder of the pot, divided by the number of tickets bought for that side.

Pool = Sum of all bets * 0.9

Payout = (wager/sum of all bets) * Pool

[BowStreetRunner]

Okay, as I understand it you want to determine the payout (2:1, 3:1, 5:2, etc.) based on how much was bet on each gladiator. (This is actually not a bad method for determining odds in situations when large numbers of bettors are involved. Based on the theory of efficient markets and market discovery, as the number of bettors increases, the accuracy of their predictions becomes greater.)

The simplest way is to take the amount bet on each gladiator divided by the total amount bet on all competitors expressed as a fraction. So 1000 + 2500 + 4500 = 8000. Therefore:

1000/8000 = 1/8
2500/8000 = 5/16
4500/8000 = 9/16

Now to convert fractions to odds is simply [a/b is (b-a) to a], thus:

1/8 = (8-1) to 1 = 7 to 1
5/16 = (16-5) to 5 = 11 to 5
9/16 = (16-9) to 9 = 7 to 9

[bookguy]

Okay, as I understand it you want to determine the payout (2:1, 3:1, 5:2, etc.) based on how much was bet on each gladiator. (This is actually not a bad method for determining odds in situations when large numbers of bettors are involved. Based on the theory of efficient markets and market discovery, as the number of bettors increases, the accuracy of their predictions becomes greater.)

The simplest way is to take the amount bet on each gladiator divided by the total amount bet on all competitors expressed as a fraction. So 1000 + 2500 + 4500 = 8000. Therefore:

1000/8000 = 1/8
2500/8000 = 5/16
4500/8000 = 9/16

Now to convert fractions to odds is simply [a/b is (b-a) to a], thus:

1/8 = (8-1) to 1 = 7 to 1
5/16 = (16-5) to 5 = 11 to 5
9/16 = (16-9) to 9 = 7 to 9


Then you change the odds slightly to favor the arena so they will make money on average. So you might change the 11 to 5 "fair payout" for C2 to a 2 to 1 actual payout. Then when C2 wins your player will get 100 gp in addition to getting back the 50 gp he wagered. In this scenario, the arena collects 5500 and pays out 5000 for a net profit of 500 gp (if the odds were not changed, they would have broken even). Of course, this is not exact because everyone makes their wagers at a different rate depending on the people who wager before them.