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# Odds

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# Odds

In common speech the terms odds and risk are used without perhaps being absolutely clear as to the meaning of these expressions. In the UK, particularly in horse racing, fractional odds are often quoted — for example, a horse is quoted at odds of 7:1, or 5:4. In the example of 7:1 (against) this means that if you place a bet of £1 and your horse wins, you will get £7 plus your original sum (or stake), i.e. £8 back. If you bet £10 initially and your horse wins you get your £10 back plus £70. The probability of winning is thus p=1/8 and of losing is q=7/8, with the odds being p/q=1/7 or 1:7 (i.e. 7:1 against). In US betting such odds would be expressed as +700 (moneyline odds).

Because such odds are expressed as simple fractions, not decimals, the odds quoted as 5:4 actually means 1.25:1, and the calculations are as before. If the odds were 2:1 on as opposed to 2:1 against, the fractional ration would be 0.5:1 and you would only get back £1 plus your stake on a £2 bet.

In health studies the term odds ratio (OR) is often used. This is simply the ratio of odds for two groups, typically the odds of becoming infected with some disease or acquiring some condition. A commonly encountered situation is the ratio of odds for those exposed to an infectious or environmental agent to those not exposed. If we denote the exposed group as group 1, and the non-exposed group as group 2 then the odds and the odds ratio can be expressed in the form of a 2x2 table of outcomes or observed frequencies:

 Exposed Not exposed Infected A C Not infected B D

From this table we see that the four odds proportions are p1=A/(A+B), p2=C/(C+D), q1=B/(A+B), and q2=D/(C+D) and thus the odds ratio (OR) can be written as:

For further discussion of odds ratios and related analysis of cross-tabulated data of this kind, see the discussions on statistics in medical research, and on the analysis of contingency tables.