Odds

When Moneyline odds are quoted as a positive number +X, it means that a wager pays X to 100.

When Moneyline odds are quoted as a negative number −X, it means that a wager pays 100 to X.

The language of odds, such as the use of phrases like "ten to one" for intuitively estimated risks, is found in the sixteenth century, well before the development of probability theory.

[1] Shakespeare wrote: Knew that we ventured on such dangerous seas That if we wrought out life 'twas ten to oneThe sixteenth-century polymath Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes.

For example, the odds that a randomly chosen day of the week is during a weekend are two to five (2:5), as days of the week form a sample space of seven outcomes, and the event occurs for two of the outcomes (Saturday and Sunday), and not for the other five.

These definitions are equivalent, since dividing both terms in the ratio by the number of outcomes yields the probabilities:

In casual use, the words odds and chances (or chance) are often used interchangeably to vaguely indicate some measure of odds or probability, though the intended meaning can be deduced by noting whether the preposition between the two numbers is to or in.

Most simply, odds are frequently multiplied or divided, and log converts multiplication to addition and division to subtractions.

However, in more variable situations, such as a multi-runner horse race or a football match between two unequally matched teams, betting "at odds" provides the possibility to take the respective likelihoods of the possible outcomes into account.

The use of odds in gambling facilitates betting on events where the probabilities of different outcomes vary.

Favoured by bookmakers in the United Kingdom and Ireland, and also common in horse racing, fractional odds quote the net total that will be paid out to the bettor, should they win, relative to the stake.

[8] Odds of 4/1 would imply that the bettor stands to make a £400 profit on a £100 stake.

However, not all fractional odds are traditionally read using the lowest common denominator.

The European odds also represent the potential winnings (net returns), but in addition they factor in the stake (e.g. 6/5 or 1.2 plus 1 = 2.2).

[10] Favoured in continental Europe, Australia, New Zealand, Canada, and Singapore, decimal odds quote the ratio of the payout amount, including the original stake, to the stake itself.

For example, if you bet €100 on Liverpool to beat Manchester City at 2.00 odds the payout, including your stake, would be €200 (€100 × 2.00).

Decimal odds are favoured by betting exchanges because they are the easiest to work with for trading, as they reflect the inverse of the probability of an outcome.

A "moneyline" wager refers to odds on the straight-up outcome of a game with no consideration to a point spread.

In most cases, the favorite will have negative moneyline odds (less payoff for a safer bet) and the underdog will have positive moneyline odds (more payoff for a risky bet).

In gambling, the odds on display do not represent the true chances (as imagined by the bookmaker) that the event will or will not occur, but are the amount that the bookmaker will pay out on a winning bet, together with the required stake.

In formulating the odds to display the bookmaker will have included a profit margin which effectively means that the payout to a successful bettor is less than that represented by the true chance of the event occurring.

This profit is known as the 'overround' on the 'book' (the 'book' refers to the old-fashioned ledger in which wagers were recorded, and is the derivation of the term 'bookmaker') and relates to the sum of the 'odds' in the following way: In a 3-horse race, for example, the true probabilities of each of the horses winning based on their relative abilities may be 50%, 40% and 10%.

In order to generate a profit on the wagers accepted, the bookmaker may decide to increase the values to 60%, 50% and 20% for the three horses, respectively.

This value of 30 represents the amount of profit for the bookmaker if he gets bets in good proportions on each of the horses.

And the expected value of his profit is positive even if everybody bets on the same horse.

The art of bookmaking is in setting the odds low enough so as to have a positive expected value of profit while keeping the odds high enough to attract customers, and at the same time attracting enough bets for each outcome to reduce his risk exposure.

[15] Making a profit in gambling involves predicting the relationship of the true probabilities to the payout odds.

They reflect the balance of wagers on either side of the event, and include the deduction of a bookmaker's brokerage fee ("vig" or vigorish).

Also, depending on how the betting is affected by jurisdiction, taxes may be involved for the bookmaker and/or the winning player.

This may be taken into account when offering the odds and/or may reduce the amount won by a player.