betting

Parimutuel betting (from the French language, Pari Mutuel or mutual betting) is a betting system in which all bets of a particular type are placed together in a pool; taxes and a house "take" or "vig" are removed, and payoff odds are calculated by sharing the pool among all winning bets. In some countries it is known as the Tote after the totalisator which calculates and displays bets already made.

The parimutuel system is used in gambling on horse racing, greyhound racing, jai alai, and all sporting events of relatively short duration in which participants finish in a ranked order. A modified parimutuel system is also used in some lottery games.
Parimutuel betting differs from fixed-odds betting in that the final payout is not determined until the pool is closed – in fixed odds betting, the payout is agreed at the time the bet is sold.

Parimutuel gambling is frequently state-regulated, and offered in many places where gambling is otherwise illegal. Parimutuel gambling is often also offered at "off track" facilities, where players may bet on the events without actually being present to observe them in person.
Consider a hypothetical event which has 8 possible outcomes, in a country using a decimal currency such as dollars. Each outcome has a certain amount of money wagered:
1 $30.00
2 $70.00
3 $12.00
4 $55.00
5 $110.00
6 $47.00
7 $150.00
8 $40.00

Thus the total pool of money on the event is $514.00. Following the start of the event, no more wagers are accepted. The event is decided and the winning outcome is determined to be Outcome 4 with $55.00 wagered. The payout is now calculated. First the commission or take for the wagering company is deducted from the pool, for example with a commission rate of 14.25% the calculation is: $514 × (1 - 0.1425) = $440.76. The remaining amount in the pool is now distributed to those who wagered on Outcome 4: $440.76 / $55 ≈ $8 per $1 wagered. This payout includes the $1 wagered plus an additional $7 profit. Thus, the odds on Outcome 4 are 7-to-1 (or, expressed as decimal odds, 8).

Often at certain times prior to the event, betting agencies will provide approximates for what should be paid out for a given outcome should no more bets be accepted at the current time. Using the wagers and commission rate above (14.25%), an approximates table in decimal odds would be:
1 $14.69
2 $6.30
3 $36.73
4 $8.01
5 $4.01
6 $9.38
7 $2.94
8 $11.02

In real-life examples such as horse racing, the pool size often extends into millions of dollars with many different types of outcomes (winning horses) and complex commission calculations.

Sometimes the amounts paid out are rounded down to a denomination interval—in the United States and Australia, 10¢ intervals are used. The rounding loss is sometimes known as breakage and is retained by the betting agency as part of the commission.

The above description of the mechanics of parimutuel wagering would suggest that it is impossible for the wagering company (the "house") to lose money, as the commission is deducted before the payouts are calculated. However, in rare circumstances, it is possible for the house to lose money on an event. This situation can occur when there are legal requirements for minimum winning payouts (for example, there may be a legal requirement to pay at least $1.10 on a winning one dollar wager). If the minimum legal winning payout exceeds the payout computed by the standard parimutuel mathematics by a sufficient amount, the house might lose money on this set of wagers.

In horse racing, a practical example of this circumstance might be when an overwhelming favorite wins. The parimutuel calculation results might call for a very small winning payout (say, $1.02 or $1.03 on a dollar bet), but the legal regulation would require a larger payout (e.g., $1.10 on a dollar bet). In North America, this condition is usually referred to as a minus pool.
Algebraic summary

In an event with a set of n possible outcomes, with wagers W1, W2, …, Wn the total pool of money on the event is

W_T = \sum^n_{i=1} W_i.

After the wagering company deducts a commission rate of r from the pool, the amount remaining to be distributed between the successful bettors is WR = WT(1 − r). Those who bet on the successful outcome m will receive a payout of WR / Wm for every dollar they bet on it.