Odds

Odds can be expressed as a ratio of two numbers, in which case it is not unique – scaling both terms by the same factor does not change the proportions: 1:1 odds and 100:100 odds are the same (even odds). Odds can also be expressed as a number, by dividing the terms in the ratio – in this case it is unique (different fractions can represent the same rational number). Odds as a ratio, odds as a number, and probability (also a number) are related by simple formulas, and similarly odds in favor and odds against, and probability of success and probability of failure have simple relations. Odds range from 0 to infinity, while probabilities range from 0 to 1, and hence are often represented as a percentage between 0% and 100%: reversing the ratio switches odds for with odds against, and similarly probability of success with probability of failure.

Given odds (in favor) as the ratio W:L (Wins:Losses), the odds in favor (as a number) ${\displaystyle o_{f}}$ and odds against (as a number) ${\displaystyle o_{a}}$ can be computed by simply dividing, and are multiplicative inverses:

${\displaystyle {\begin{aligned}o_{f}&=W/L=1/o_{a}\\o_{a}&=L/W=1/o_{f}\\o_{f}\cdot o_{a}&=1\end{aligned}}}$

Analogously, given odds as a ratio, the probability of success or failure can be computed by dividing, and the probability of success and probability of failure sum to unity (one), as they are the only possible outcomes. In case of a finite number of equally likely outcomes, this can be interpreted as the number of outcomes where the event occurs divided by the total number of events:

${\displaystyle {\begin{aligned}p&=W/(W+L)=1-q\\q&=L/(W+L)=1-p\\p+q&=1\end{aligned}}}$

Given a probability p, the odds as a ratio is ${\displaystyle p:q}$ (probability of success to probability of failure), and the odds as numbers can be computed by dividing:

${\displaystyle {\begin{aligned}o_{f}&=p/q=p/(1-p)=(1-q)/q\\o_{a}&=q/p=(1-p)/p=q/(1-q)\end{aligned}}}$

Conversely, given the odds as a number ${\displaystyle o_{f},}$ this can be represented as the ratio ${\displaystyle o_{f}:1,}$ or conversely ${\displaystyle 1:(1/o_{f})=1:o_{a},}$ from which the probability of success or failure can be computed:

${\displaystyle {\begin{aligned}p&=o_{f}/(o_{f}+1)=1/(o_{a}+1)\\q&=o_{a}/(o_{a}+1)=1/(o_{f}+1)\end{aligned}}}$

Thus if expressed as a fraction with a numerator of 1, probability and odds differ by exactly 1 in the denominator: a probability of 1 in 100 (1/100 = 1%) is the same as odds of 1 to 99 (1/99 = 0.0101... = 0.01), while odds of 1 to 100 (1/100 = 0.01) is the same as a probability of 1 in 101 (1/101 = 0.00990099... = 0.0099). This is a minor difference if the probability is small (close to zero, or "long odds"), but is a major difference if the probability is large (close to one).

These are worked out for some simple odds:

odds (ratio)	${\displaystyle o_{f}}$	${\displaystyle o_{a}}$	${\displaystyle p}$	${\displaystyle q}$
1:1	1	1	50%	50%
0:1	0	∞	0%	100%
1:0	∞	0	100%	0%
2:1	2	0.5	67%	33%
1:2	0.5	2	33%	67%
4:1	4	0.25	80%	20%
1:4	0.25	4	20%	80%
9:1	9	0.1	90%	10%
10:1	10	0.1	90.90%	9.09%
99:1	99	0.01	99%	1%
100:1	100	0.01	99.0099%	0.9900%

These transforms have certain special geometric properties: the conversions between odds for and odds against (resp. probability of success with probability of failure) and between odds and probability are all Möbius transformations (fractional linear transformations). They are thus specified by three points (sharply 3-transitive). Swapping odds for and odds against swaps 0 and infinity, fixing 1, while swapping probability of success with probability of failure swaps 0 and 1, fixing .5; these are both order 2, hence circular transforms. Converting odds to probability fixes 0, sends infinity to 1, and sends 1 to .5 (even odds are 50% likely), and conversely; this is a parabolic transform.

In probability theory and statistics, odds and similar ratios may be more natural or more convenient than probabilities. In some cases the log-odds are used, which is the logit of the probability. Most simply, odds are frequently multiplied or divided, and log converts multiplication to addition and division to subtractions. This is particularly important in the logistic model, in which the log-odds of the target variable are a linear combination of the observed variables.

Similar ratios are used elsewhere in statistics; of central importance is the likelihood ratio in likelihoodist statistics, which is used in Bayesian statistics as the Bayes factor.

Odds are particularly useful in problems of sequential decision making, as for instance in problems of how to stop (online) on a last specific event which is solved by the odds algorithm.

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: 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 16 times higher on the less likely event (4–1 against, or 4) than on the more likely one (1–4, or 4–1 on, or 0.25).

Example #1

There are 5 pink marbles, 2 blue marbles, and 8 purple marbles. What are the odds in favor of picking a blue marble?

Answer: The odds in favour of a blue marble are 2:13. One can equivalently say, that the odds are 13:2 against. There are 2 out of 15 chances in favour of blue, 13 out of 15 against blue.

In probability theory and statistics, where the variable p is the probability in favor of a binary event, and the probability against the event is therefore 1-p, "the odds" of the event are the quotient of the two, or ${\displaystyle {\frac {p}{1-p}}}$. That value may be regarded as the relative probability the event will happen, expressed as a fraction (if it is less than 1), or a multiple (if it is equal to or greater than one) of the likelihood that the event will not happen.

In the first example at top, saying the odds of a Sunday are "one to six" or, less commonly, "one-sixth" means the probability of picking a Sunday randomly is one-sixth the probability of not picking a Sunday. While the mathematical probability of an event has a value in the range from zero to one, "the odds" in favor of that same event lie between zero and infinity. The odds against the event with probability given as p are ${\displaystyle {\frac {1-p}{p}}}$. The odds against Sunday are 6:1 or  6/1 = 6. It is 6 times as likely that a random day is not a Sunday.

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 he or she win, relative to the stake.[9] Odds of 4/1 would imply that the bettor stands to make a £400 profit on a £100 stake. If the odds are 1/4, the bettor will make £25 on a £100 stake. In either case, having won, the bettor always receives the original stake back; so if the odds are 4/1 the bettor receives a total of £500 (£400 plus the original £100). Odds of 1/1 are known as evens or even money.

The numerator and denominator of fractional odds are always integers, thus if the bookmaker's payout was to be £1.25 for every £1 stake, this would be equivalent to £5 for every £4 staked, and the odds would therefore be expressed as 5/4. However, not all fractional odds are traditionally read using the lowest common denominator. For example, given that there is a pattern of odds of 5/4, 7/4, 9/4 and so on, odds which are mathematically 3/2 are more easily compared if expressed in the equivalent form 6/4.

Fractional odds are also known as British odds, UK odds,[10] or, in that country, traditional odds. They are typically represented with a "/" but can also be represented with a "-", e.g. 4/1 or 4-1. Odds with a denominator of 1 are often presented in listings as the numerator only.

A variation of fractional odds is known as Hong Kong odds. Fractional and Hong Kong odds are actually exchangeable. The only difference is that the UK odds are presented as a fractional notation (e.g. 6/5) whilst the Hong Kong odds are decimal (e.g. 1.2). Both exhibit the net return.

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).[11]

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. Therefore, the decimal odds of an outcome are equivalent to the decimal value of the fractional odds plus one.[12] Thus even odds 1/1 are quoted in decimal odds as 2.00. The 4/1 fractional odds discussed above are quoted as 5.00, 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. When looking at decimal odds in betting terms, the underdog has the higher of the two decimals, while the favorite has the lower of the two. To calculate decimal odds, you can use the equation Return = Initial Wager x Decimal Value.[13] For example, if you bet €100 on Liverpool to beat Manchester City at 2.00 odds you would win €200 (€100 x 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.[14] For example, a quoted odds of 5.00 equals to a probability of 1 / 5.00, that is 0.20 or 20%.

Decimal odds are also known as European odds, digital odds or continental odds.[10]

Moneyline odds are favoured by American bookmakers. The figure quoted is either positive or negative.

• When moneyline odds are positive, the figure indicates how much money will be won on a $100 wager (this is done for an outcome that is considered less likely to happen than not). For example, a net payout of 4/1 would be quoted as +400.
• When moneyline odds are negative, the figure indicates how much money must be wagered to win $100 (this is done for an outcome that is considered more likely to happen than not). For example, a net payout of 1/4 would be quoted as -400.

Moneyline odds are often referred to as American odds. 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). However, if the teams are evenly matched, both teams can have a negative line at the same time (e.g.,. -110 -110 or -105 -115), due to house take.

Wholesale odds are the "real odds" or 100% probability of an event occurring. This 100% book is displayed without any bookmaker's profit margin, often referred to as a bookmaker's "overround" built in.

A "wholesale odds" index is an index of all the prices in a probabilistic market operating at 100% competitiveness and displayed without any profit margin factored for market participants.