In some games of chance, this is also the most convenient way for a person to understand how much winnings will be paid if the selection is successful: the person will be paid 'six' of whatever stake unit was bet for each 'one' of the stake unit wagered. For example, a £10 winning bet at 6/1 will win '6 × £10 = £60' with the original £10 stake also being returned.
Taking an event with a 1 in 5 probability of occurring (i.e. a probability of 1/5, 0.2 or 20%), then the odds are 0.2 / (1 − 0.2) = 0.2 / 0.8 = 0.25. This figure (0.25) represents the stake necessary for a person to win one unit on a successful wager. This may be scaled up by any convenient factor to give whole number values. E.g. If a stake of 0.25 wins 1 unit, then scaling by a factor of four means a stake of 1 wins 4 units. If you bet 1 at these odds and the event occurred, you would receive back 4 plus your original 1 stake. This would be presented in fractional odds of 4 to 1 against (written as 4-1, 4:1, or 4/1), in decimal odds as 5.0 to include the returned stake, in craps payout as 5 for 1, and in moneyline odds as +400 representing the gain from a 100 stake.
By contrast, for an event with a 4 in 5 probability of occurring (i.e. a probability of 4/5, 0.8 or 80%), then the odds are 0.8 / (1 − 0.8) = 4. If you bet 4 at these odds and the event occurred, you would receive back 1 plus your original 4 stake. This would be presented in fractional odds of 4 to 1 on (written as 1/4 or 1-4), in decimal odds as 1.25 to include the returned stake, in craps as 5 for 4, and in moneyline odds as −400 representing the stake necessary to gain 100.
In a 3-horse race, for example, the true chances of each of the horses winning based on their relative abilities may be 50%, 40% and 10%. These are the relative probabilities of the horses winning and are simply the bookmaker's 'odds' multiplied by 100 for convenience. The total of these three percentages is 100, thus representing a fair 'book'. The true odds of winning for each of the three horses is evens, 6-4 and 9-1 respectively. In order to generate a profit on the wagers accepted by the bookmaker he may decide to increase the values to 60%, 50% and 20% for the three horses, representing odds of 4-6, Evens and 4-1. These values now total 130, meaning that the book has an overround of 30 (130 − 100). This value of 30 represents the amount of profit for the bookmaker if he accepts bets in the correct proportions on each of the horses. The art of bookmaking is that he will take in, for example, $130 in wagers and only pay $100 back (including stakes) no matter which horse wins.
Profiting in gambling involves predicting the relationship of the true probabilities to the payout odds. If you can consistently make bets where the odds of paying out are better (pay out more) than the true odds of the event, then over time (in theory) you will come out ahead.
The odds or amounts the bookmaker will pay are determined by the amounts bet on each of the respective possible events. 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).
In popular parlance surrounding uncertain events, the expression "better than even" usually implies a better than (greater than) 50% chance of the event occurring, which is exactly the opposite of the meaning of the expression when used in a gaming context.
The odds are a ratio of probabilities; an odds ratio is a ratio of odds, that is, a ratio of ratios of probabilities. Odds-ratios are often used in analysis of clinical trials. While they have useful mathematical properties, they can produce counter-intuitive results: in the example above an event with an 80% probability of occurring is four times more likely to happen than an event with a 20% probability, but the odds are actually 16 times higher on the less likely event (4-1 against) than on the more likely one (1-4, or 4-1 on).