for: Odds (band) In probability theory and statistics the odds in favour of an event or a proposition are the quantity p / (1  p), where p is the probability of the event or proposition. The odds against the same event are (1  p) / p. For example, if you chose a random day of the week, then the odds that you would choose a Sunday would be 1/6, not 1/7. The odds against you choosing Sunday are 6/1. These 'odds' are actually relative probabilities. Generally, 'odds' are not quoted to the general public in this format because of the natural confusion with the chance of an event occurring being expressed fractionally as a probability. Thus, the probability of choosing Sunday at random from the days of the week is 'one-seventh' (1/7), and although a bookmaker may (for his own purposes) use 'odds' of 'one-sixth' the overwhelming everyday use by most people is odds of the form 6 to 1, 6-1, or 6/1 (all read as 'six-to-one') where the first figure represents the number of ways of failing to achieve the outcome and the second figure is the number of ways of achieving a favourable outcome: thus these are "odds against". In other words, an event with m to n "odds against" would have probability n/(m + n), while an event with m to n "odds on" would have probability m/(m + n).

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 × Â£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.