Odds to Probability Converter

A probability is a part over the whole: $P = \dfrac{\text{favourable}}{\text{total}}$, always between 0 and 1. Odds are a part over the rest: a ratio of two counts that never includes the total.

• Odds in favour $a:b$ means $a$ successes for every $b$ failures, so $P = \dfrac{a}{a+b}$.
• Odds against $a:b$ (the usual reading of "7 to 3" and of UK fractional odds) means $a$ failures per $b$ successes, so $P = \dfrac{b}{a+b}$.

Going the other way, from a probability $p$:

$\text{odds in favour} = \frac{p}{1-p}, \qquad \text{odds against} = \frac{1-p}{p}$

The assumption that matters: these formulas are exact only when the ratio counts equally likely outcomes, or is stated as a true ratio of probabilities. A quoted market price is not that — see the overround below.

Decimal (European) — total return per unit staked, so the implied probability is simply

$p = \frac{1}{d}$

Fractional (UK) — $a/b$ is odds against: $p = \dfrac{b}{a+b}$. So $5/1$ gives $p = 1/6 \approx 0.167$.

American (moneyline) — two rules, split at even money:

$p = \frac{100}{m + 100} \;\; (m > 0), \qquad p = \frac{|m|}{|m| + 100} \;\; (m < 0)$

Positive lines are underdogs ($p < 0.5$); negative lines are favourites ($p > 0.5$). A line of $+100$ or $-100$ is exactly $p = 0.5$.

Overround. Add the implied probabilities of every outcome in a market. Two sides at $-110$ give $\frac{110}{210} = 0.5238$ each, totalling $1.0476$ — a 4.76% excess. Because the total exceeds 1, an implied probability is a price, not an estimate of the true chance; divide each by the total to normalise.

• Reading a ratio in the wrong direction. "3 to 1" almost always means odds against, i.e. $p = 1/4$, not $3/4$. Fractional odds are always against.
• Dividing by the total. Odds $a:b$ has denominator $a+b$ for the probability but the odds themselves are $a$ over $b$. Mixing the two gives $p/(1-p)$ where you wanted $p$.
• Using the positive American formula on a negative line. $-220$ is not $100/320$; it is $220/320 = 68.75\%$.
• Treating implied probability as a true probability. The overround makes every quoted implied probability too high.
• Adding odds. Odds do not add or average. Convert to probabilities, combine those, then convert back.
• Percent to odds without converting first. From 40%, odds in favour are $0.40/0.60 = 2:3$, not $40:100$ reduced carelessly.