Odds to probability:
what 3:1, 2 to 5 and 2.50 really mean
Odds and probability describe the same uncertainty in different languages. Mixing them up is the most common reason a chance sounds better than it is.
Calcylator Editorial Team
Updated · 5 min read
Two ways to say the same chance
Probability compares the event with everything that could happen: 1 chance in 4 is 25%. Odds compare the event with its opposite: 1 chance for the event against 3 chances for something else. The two always describe one underlying chance, but their numbers are not interchangeable.
An event with probability 0.25 has odds of 1 to 3 in favour, which is the same as 3 to 1 against. If you read '3 to 1' as 'three out of one' or as 75%, you have flipped the story. Probability can never exceed 1 (100%), while odds can run to any size, which is part of why they feel less intuitive.
Odds are the usual language in horse racing and betting, in some medical studies (as an odds ratio), and in everyday speech like 'the odds are against it'. Probability is the language of most mathematics and forecasts.
The conversion rules
- a:
- chances for the event
- b:
- chances against the event
- a:
- chances against
- b:
- chances for
- p:
- probability as a decimal between 0 and 1
- decimal odds:
- payout per unit staked including the stake, e.g. 2.50
Fractional odds such as 5/2 are quoted in the 'against' direction in most betting markets: 5/2 means 5 to 2 against, so the implied probability is 2 ÷ (5 + 2) = 28.6%. Always check which direction a source uses.
Worked conversions
Odds in favour
3 to 2
Total parts
3 + 2 = 5
Probability
3 ÷ 5
Probability
60%
Odds against
3 to 1
Total parts
3 + 1 = 4
Probability of the event
1 ÷ 4
Probability
25%
The chance it does not happen is 3 ÷ 4 = 75%.
Decimal odds
2.50
Implied probability
1 ÷ 2.50
Probability
40%
Equivalent to odds of 3 to 2 against.
Going the other way, a 20% chance is p ÷ (1 − p) = 0.20 ÷ 0.80 = 0.25, which is 1 to 4 in favour or 4 to 1 against. A 75% chance is 0.75 ÷ 0.25 = 3, so 3 to 1 in favour.
Combining chances: why multiplying beats adding odds
Odds cannot be added or multiplied directly; probabilities can. To find the chance of two independent events both happening, convert to probabilities, multiply, and convert back if odds are needed.
Each event
3 to 1 against, so 25% (0.25)
Both happen
0.25 × 0.25 = 0.0625
Back to odds
1 ÷ 16 → 15 to 1 against
Chance of both
6.25%
Not '6 to 1' and not 12.5%; the multiplication is done on probabilities
The mistake of multiplying the odds figures (3 × 3 = 9 to 1) understates how unlikely the pair is. The chance of at least one of two independent events is 1 − (0.75 × 0.75) = 43.75%, again worked in probabilities.
A quick reference table
| Probability | Odds in favour | Odds against | Decimal odds |
|---|---|---|---|
| 10% | 1 : 9 | 9 : 1 | 10.00 |
| 20% | 1 : 4 | 4 : 1 | 5.00 |
| 25% | 1 : 3 | 3 : 1 | 4.00 |
| 33.3% | 1 : 2 | 2 : 1 | 3.00 |
| 50% | 1 : 1 | 1 : 1 | 2.00 |
| 60% | 3 : 2 | 2 : 3 | 1.67 |
| 75% | 3 : 1 | 1 : 3 | 1.33 |
Decimal odds shown here are the 'fair' ones, with no margin built in. Real quoted odds are lower than the fair figure because the bookmaker takes a cut, which the next section illustrates.
'One in N' risks and their odds
Risk statements are often phrased as 'one in N'. That is a probability of 1/N, with odds of 1 to (N − 1) in favour. The conversion is quick enough to do on the spot.
| Statement | Probability | Odds in favour | Odds against |
|---|---|---|---|
| 1 in 20 | 5% | 1 : 19 | 19 : 1 |
| 1 in 100 | 1% | 1 : 99 | 99 : 1 |
| 1 in 1,000 | 0.1% | 1 : 999 | 999 : 1 |
| 1 in 10,000 | 0.01% | 1 : 9,999 | 9,999 : 1 |
For rare events, odds and probability are nearly equal in size: 1 in 1,000 is 1 to 999, which most people round to 1 to 1,000. The difference becomes important when the chance is large. A 90% chance is 9 to 1 in favour; saying 'nine times out of ten' and 'nine to one' is fine, but 'nine to ten' would be wrong.
In medicine, an odds ratio of 2 does not mean the risk doubled. If the baseline chance is 10% (odds 0.111), doubling the odds gives 0.222, which is a probability of 18.2%, not 20%. The gap grows as the baseline rises.
Odds ratios, and a last check on any conversion
In health research an odds ratio compares the odds of an outcome in two groups. If the odds of recovery are 3 to 1 in one group and 1 to 1 in another, the odds ratio is 3. Odds ratios are not the same as risk ratios, and they overstate the apparent effect when the outcome is common.
Logistic regression, used in credit scoring and medical risk models, works on the log of the odds because that scale stretches the 0–1 probability range across the whole number line. The jargon is heavy, but the underlying conversion is the one in this article.
A final sanity check for any conversion: the probability must land between 0 and 1, and the probabilities of all mutually exclusive outcomes must add to 1. If they do not, check which direction the odds were quoted.
When the numbers look odd, run three quick checks. First, note the direction: are the odds for the event or against it? Second, add the two parts for the denominator, whichever way round they are written. Third, confirm the answer is between 0 and 100% and that the event and its opposite add to 100%.
Common questions
How do you convert odds to probability?
For odds in favour of a to b, probability = a ÷ (a + b). For odds against a to b, probability = b ÷ (a + b). So 3 to 1 against is 1 ÷ 4 = 25%, and 3 to 2 in favour is 3 ÷ 5 = 60%.
What is the difference between odds and probability?
Probability compares an event with all possible outcomes, while odds compare it with the opposite outcome. A 25% probability is 1 to 3 odds in favour. Probability never exceeds 100%, but odds can be any ratio.
How do I turn decimal odds into a percentage?
Divide 1 by the decimal odds and multiply by 100. Decimal odds of 2.50 give 1 ÷ 2.50 = 0.40, or 40%. The figure includes the operator's margin, so the fair chance is slightly lower.
What does 5/1 odds mean as a probability?
Fractional 5/1 is conventionally read as 5 to 1 against, so the probability is 1 ÷ (5 + 1) = 16.7%. The decimal equivalent is 6.00. Confirm the quoting direction, since some contexts use odds in favour.
Can odds be converted to a probability above 100%?
No. A correct conversion always lands between 0% and 100%. If you get more than 100%, odds were read in the wrong direction or the numbers were added up wrongly in the denominator.
Was this guide helpful?
Continue reading
View all blogsSample Size for a Survey: How Many Responses?
A 95% confidence level and a ±5% margin need about 385 responses for a large population. See the formula and how a smaller population shrinks that.
4 min read
T-Test Statistic Formula With a Worked Example
The one-sample t statistic is (mean − claimed mean) ÷ (SD ÷ √n). With mean 54, claim 50, SD 10 and n = 25, t = 2.0 on 24 degrees of freedom.
5 min read
Z-Score Formula: Standard Deviations From the Mean
A z-score subtracts the mean and divides by the standard deviation: (85 − 70) ÷ 10 = 1.5. That puts the value about the 93rd percentile if scores are normal.
5 min read
