Odds vs Probability: How to Convert Between Them
People use "odds" and "probability" as if they were interchangeable, but they are different scales. Probability compares the outcome you want with all outcomes. Odds compare it with the outcomes you do not want. Mixing them up is one of the easiest ways to misread a statistic.
This guide gives the two conversion formulas, a reference table, the link to betting formats, and an example showing why an odds ratio can look much bigger than the risk ratio it is often confused with.
Two ways to describe the same chance
Rolling a six on a fair die has a probability of 1 in 6, or about 16.7%. The odds in favour are 1 to 5, because there is one way to win and five ways not to. The odds against are the reverse, 5 to 1.
odds = p ÷ (1 − p) p = odds ÷ (1 + odds)Odds written as a single number, such as 3 for 3 to 1. Probability p is a value between 0 and 1.
Worked example: a probability of 0.75 gives odds of 0.75 ÷ 0.25 = 3, which is 3 to 1 in favour. Going the other way, odds of 3 to 2 are 1.5 as a single number, so p = 1.5 ÷ 2.5 = 0.6. Equivalently, p = 3 ÷ (3 + 2) = 0.6.
Probability is capped at 1 (certainty), while odds have no upper limit. That is one reason odds are popular in some branches of statistics: they stretch out the high end of the scale.
A quick conversion table
| Probability | Odds in favour | Odds against |
|---|---|---|
| 10% | 1 to 9 | 9 to 1 |
| 20% | 1 to 4 | 4 to 1 |
| 25% | 1 to 3 | 3 to 1 |
| 50% | 1 to 1 | 1 to 1 |
| 75% | 3 to 1 | 1 to 3 |
| 90% | 9 to 1 | 1 to 9 |
Notice that 50% is the only point where the odds in favour and against match. The odds calculator converts between these forms for any input.
The wording trap: "1 in 4" versus "1 to 4"
A chance of "1 in 4" is a probability of 1 ÷ 4 = 25%. But "odds of 1 to 4" mean one favourable outcome for every four unfavourable ones, which is 1 ÷ 5 = 20%. Both phrases sound alike and give different numbers.
When you read a figure, check whether the second number is the total ("in") or the unfavourable count ("to"). If the source is unclear, work backwards from any raw counts it gives.
A handy shortcut follows from this: a chance of 1 in n equals odds of 1 to (n − 1). So 1 in 10 is 1 to 9, 1 in 100 is 1 to 99, and 1 in 2 is 1 to 1. For small probabilities the two scales are nearly identical, which is why the confusion rarely matters for rare events and matters a great deal for common ones.
How betting odds map to probability
Betting markets quote odds in a few formats. A fractional price of 5/1 pays 5 in profit for each 1 staked, and its decimal equivalent, which includes the returned stake, is 6.00. The probability implied by a decimal price is simply 1 divided by it.
implied probability = 1 ÷ decimal odds
| Fractional | Decimal | Implied probability |
|---|---|---|
| 1/2 | 1.50 | 66.7% |
| 1/1 | 2.00 | 50.0% |
| 2/1 | 3.00 | 33.3% |
| 5/1 | 6.00 | 16.7% |
| 10/1 | 11.00 | 9.1% |
A price of 6.00 implies 1 ÷ 6 = 16.7%. Now take a two-way market where both outcomes are priced at 1.90. Each implies 1 ÷ 1.90 = 52.6%, and the two add up to 105.3%. The 5.3% above 100% is the bookmaker's built-in margin, so the prices are slightly worse than the true odds would be.
This is arithmetic, not betting or financial advice. Prices that imply a probability above the true one lose money on average over time.
Odds ratios versus risk ratios
Studies often report an odds ratio, but readers tend to interpret it as a risk ratio (relative risk). The two agree closely when the outcome is rare, say under about 10% in both groups, and drift apart when it is common. The odds ratio is always further from 1 than the risk ratio, so quoting it as a risk ratio exaggerates the effect.
Worked example: 40 of 100 people in group A have an outcome, against 20 of 100 in group B. The risk ratio is 0.40 ÷ 0.20 = 2.0. The odds are 0.40 ÷ 0.60 = 0.667 for A and 0.20 ÷ 0.80 = 0.25 for B, so the odds ratio is 0.667 ÷ 0.25 = 2.67.
Saying the first group is "2.67 times as likely" would overstate the difference. Compute each with the odds ratio calculator and the relative risk calculator, and quote the one that matches your meaning.
- Odds Calculator: Enter odds in any one format and see them in all the others, together with the implied probability. Add a stake to see what you would win and what you would be paid.
- Odds Ratio Calculator: Enter the number of events and non-events in an exposed group and a control group to get the odds ratio, its confidence intervals at the level you choose, and a p-value for the null of no association.
- Relative Risk Calculator: Enter how many people in an exposed group and in a control group did and did not experience the outcome. You get the risk ratio, two-sided and one-sided confidence intervals, a one-sided p-value, the z-score and the number needed to treat.
This article is general information, not professional advice. See our Disclaimer and Editorial Policy.
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