odds

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 × £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 100In gambling, the odds on display do not represent the true chances that the event will occur, but are the amounts that the bookmaker will pay out on winning bets. In formulating his odds to display the bookmaker will have included a profit margin which effectively means that the payout to a successful punter is less than that represented by the true chance of the event occurring. This profit is known as the 'over-round' on the 'book' (the 'book' relates to the old-fashioned ledger that wagers were recorded in and thus gives us the term 'bookmaker') and relates to the sum of the 'odds' in the following way:
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 vigorishThe terms 'even odds', 'even money' or simply 'Evens' imply that the payout will be 'one-for-one' or 'double-your-money'. Assuming there is no bookmaker’s fee or built-in profit margin, this means that the actual probability of winning is 50%. The term “better than even odds” looks at it from the perspective of a gambler rather than a statistician. If the odds are Evens (1-1), and you bet 10, you would win 10. If the gamble was paying 4-1 and the event occurred, you would make a profit of 40. So, it is better than Evens from the gambler’s perspective because it pays out more than one-for-one. If an event is more positively favored to occur than a 50-50 chance then the odds will be worse than Evens, and the bookmaker will pay out less than one-for-one.
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).
The logarithm of the odds is the logit of the probability.
It is customary with fixed-odds gambling to know the odds at the time of the placement of the wager (the "live price"), although this category also includes wagers whose price is determined only when the race or game starts (the "starting prices"). It is ideal for a bookmaker to price up a book such that the net outcome will always be in his favour, i.e. the sum of the probabilities quoted for all possible outcomes will be in excess of 100%. The excess over 100% (or overround) represents profit to the bookmaker in the event of a balanced book. In the more usual case of an imbalanced book, the bookmaker may have to pay out more winnings than what is staked, or he may earn more than mathematically expected. An imbalanced book may arise since there is no way for a bookmaker either to know the true probabilities for the outcome of competitions left to human effort or to predict the bets that will be attracted from others by fixed odds compiled on the basis of his own personal view and knowledge.
With the advent of Internet and bet exchange betting, the possibility of fixed-odds arbitrage actions and Dutch books against bookmakers and exchanges has expanded significantly. Betting exchanges in particular act like a stock exchange, allowing the odds to be set in the course of trading between individual bettors, usually leading to quoted odds that are reasonably close to the "true odds."
In making a bet where your expected value is positive, you are said to be getting "the best of it". For example, if you were to bet $1 at 10 to 1 odds (you could win $10) on the outcome of a coin flip, you would be getting "the best of it" and you should always make the bet. However if someone offered you odds of 10 to 1 that a card chosen at random from a regular 52 card deck would be the ace of spades, then you would be getting "the worst of it" because the chance is only 1 in 52 that the ace will be chosen. It is mathematically disadvantageous to make a bet where you are getting "the worst of it."
When making a bet where you must put more at stake than you stand to win, you are laying the odds or laying the bet. So, for example, if you bet $1000 that it will rain tomorrow, and if you win you will only win $200 but if you lose you will lose your entire $1000, then you are laying a bet. It is possible that you could be getting "the best of it" or "the worst of it" when you lay a bet; the fact that you are laying a bet does not necessarily mean you are getting "the worst of it". A lay bet is a bet that something won't happen, so if you lay $50 on a horse then you are betting the horse won't win.
Favoured by bookmakers in the United Kingdom and Ireland, fractional odds quote the net total that will be paid out to the bettor, should he win, relative to his stake. Odds of 4/1 ("four-to-one" or less commonly "four-to-one against") would imply that the bettor stands to make a £400 profit on a £100 stake. If the odds are 1/4 (read "one-to-four", or "four-to-one on"), the bettor will make £25 on a £100 stake. Should he win, the bettor always receives his original stake back, so if the odds are 4/1 you would actually receive a total of £500 in return (£400 plus the original £100). Odds of 1/1 are known as evens or even money. Unusually, odds of 10/3 is read as "one-hundred-to-thirty".
Fractional odds are also known as British odds, UK odds or in that country, traditional odds.
Favoured in continental Europe, Australia and Canada, decimal odds differ from fractional odds in that the bettor must first part with their stake in order to make a bet, the figure quoted is the winning amount that would be paid out to the bettor. Therefore, the decimal odds of an outcome are equivalent to the decimal value of the fractional odds plus one. Thus even odds 1/1 are quoted in decimal odds as 2. The 4/1 fractional odds discussed above are quoted as 5, while the 1/4 odds are quoted as 1.25. This is considered to be ideal for parlay betting, because the odds to be paid out are simply the product of the odds for each outcome wagered on. Decimal odds are also favoured by betting exchanges because they are the easiest to work with for trading.
Decimal odds are also known as European odds, or continental odds in the UK.
Moneyline odds are favoured by American bookmakers. There are two possibilities, the figure quote can be either positive or negative.
Moneyline odds are often referred to as American odds. Moneyline refers to odds on the straight-up outcome of a game with no consideration to a point spread.
Positive figures If the figure quoted is positive, the odds are quoting how much money will be won on a $100 wager (this is done if the odds are better than even). Even odds are quoted as $100 . Fractional odds of 4/1 would be quoted as $400, while fractional odds of 1/4 cannot be quoted as a positive figure. Negative figures If the figure quoted is negative, then the moneyline odds are quoting how much money must be wagered to win $100 (this is done if the odds are worse than even). Even odds are quoted as -$100. Fractional odds of 1/4 would be quoted as -$400, however fractional odds of 4/1 cannot be quoted as a negative figure.