Editing Odds (section)

===Mathematical relations===
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 [[Fraction (mathematics)|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 (number of outcomes that are wins:number of outcomes that are losses), the odds in favor (as a number) <math>o_f</math> and odds against (as a number) <math>o_a</math> can be computed by simply dividing, and are [[multiplicative inverse]]s:
:<math>
\begin{align}
o_f &= W/L = 1/o_a\\
o_a &= L/W = 1/o_f\\
o_f \cdot o_a &= 1
\end{align}
</math>

Analogously, given odds as a ratio, the probability of success {{mvar|p}} or failure {{mvar|q}} can be computed by dividing, and the probability of success and probability of failure sum to [[Unity (mathematics)|unity]] (one), as they are the only possible outcomes. In case of a finite number of equally probable outcomes, this can be interpreted as the number of outcomes where the event occurs divided by the total number of events:
:<math>
\begin{align}
p &= W/(W+L) = 1 - q\\
q &= L/(W+L) = 1 - p\\
p + q &= 1
\end{align}
</math>

Given a probability ''p,'' the odds as a ratio is <math>p:q</math> (probability of success to probability of failure), and the odds as numbers can be computed by dividing:
:<math>
\begin{align}
o_f &= p/q = p/(1-p) = (1-q)/q\\
o_a &= q/p = (1-p)/p = q/(1-q)
\end{align}
</math>

Conversely, given the odds as a number <math>o_f,</math> this can be represented as the ratio <math>o_f:1,</math> or conversely <math>1:(1/o_f) = 1:o_a,</math> from which the probability of success or failure can be computed:
:<math>
\begin{align}
p &= o_f/(o_f+1) = 1/(o_a + 1)\\
q &= o_a/(o_a+1) = 1/(o_f + 1)
\end{align}
</math>
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.{{overline|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.{{overline|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:
{| class=wikitable
! odds (ratio) || <math>o_f</math> || <math>o_a</math> || <math>p</math> || <math>q</math>
|-
|1:1 || 1 || 1 || 50% || 50%
|-
|0:1 || 0 || ''∞'' || 0% || 100%
|-
|1:0 || ''∞'' || 0 || 100% || 0%
|-
|2:1 || 2 || 0.5 || 66.{{overline|66}}% || 33.{{overline|33}}%
|-
|1:2 || 0.5 || 2 || 33.{{overline|33}}% || 66.{{overline|66}}%
|-
|4:1 || 4 || 0.25 || 80% || 20%
|-
|1:4 || 0.25 || 4 || 20% || 80%
|-
|colspan=5|
|-
|9:1 || 9 || 0.{{overline|1}} || 90% || 10%
|-
|10:1 || 10 || 0.1 || 90.{{overline|90}}% || 9.{{overline|09}}%
|-
|99:1 || 99 || 0.{{overline|01}} || 99% || 1%
|-
|100:1 || 100 || 0.01 || 99.{{overline|0099}}% || 0.{{overline|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 transformation]]s (fractional linear transformations). They are thus [[Möbius transformation#Specifying a transformation by three points|specified by three points]] ([[Sharply multiply transitive|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 transform]]s. Converting odds to probability fixes 0, sends infinity to 1, and sends 1 to .5 (even odds are 50% probable), and conversely; this is a [[parabolic transform]].