Two descriptions of the same chance
Probability and odds both describe how likely something is, but they are built differently.
- Probability compares the favourable outcomes with all outcomes. It is a fraction, decimal or percentage between 0 and 1 (or 0% and 100%).
- Odds compare the favourable outcomes with the unfavourable ones. They are written as a ratio such as 1 to 3 or 1:3.
A fair die gives a good example. The chance of rolling a 3 is 1 out of 6, so the probability is 1/6, about 16.7%. There is one way to get a 3 and five ways not to, so the odds in favour are 1 to 5.
Odds in favour and odds against
Odds can point in either direction.
- Odds in favour of an event: favourable outcomes to unfavourable outcomes. For rolling a 3, that is 1 to 5.
- Odds against an event: unfavourable outcomes to favourable outcomes. For rolling a 3, that is 5 to 1.
Always check which direction is being used. “Five to one” sounds like a good chance until you notice it is the odds against. A source that does not say which way it means leaves you with an ambiguous number.
From odds to probability
If the odds in favour are a to b, the probability is:
probability = a ÷ (a + b)
The denominator is the total number of outcomes, which is the favourable part plus the unfavourable part.
Examples:
- Odds in favour of 1 to 3: probability = 1 ÷ (1 + 3) = 1/4 = 25%.
- Odds in favour of 3 to 2: probability = 3 ÷ (3 + 2) = 3/5 = 60%.
- Odds against of 4 to 1: this is the same as odds in favour of 1 to 4, so probability = 1 ÷ (1 + 4) = 1/5 = 20%.
- Odds in favour of 1 to 1 (often called even odds): probability = 1 ÷ 2 = 50%.
From probability to odds
If the probability of an event is p, then the probability of it not happening is 1 − p. The odds in favour are:
odds in favour = p to (1 − p)
It is neat to work with fractions. If p = 2/5, the probability of not happening is 3/5, so the odds in favour are 2/5 to 3/5, which is the same as 2 to 3.
More examples:
| Probability | Odds in favour | Odds against |
|---|---|---|
| 1/2 (50%) | 1 to 1 | 1 to 1 |
| 1/4 (25%) | 1 to 3 | 3 to 1 |
| 1/6 (about 16.7%) | 1 to 5 | 5 to 1 |
| 3/4 (75%) | 3 to 1 | 1 to 3 |
| 1/13 (about 7.7%) | 1 to 12 | 12 to 1 |
The last row is the chance of drawing an ace from a full deck: 4 aces out of 52 cards is 1/13, and the odds in favour are 4 to 48, which simplifies to 1 to 12.
When the probability is given as a percentage, convert it first. For 30%, write it as 30/100 and the opposite as 70/100, so the odds in favour are 30 to 70, which simplifies to 3 to 7.
A quick sanity check
Two checks catch most errors.
- If the odds are balanced (equal numbers on each side), the probability should be 50%.
- A probability can never exceed 100%. If your conversion gives a bigger number, you have probably used the wrong side of the ratio. For example, odds of 3 to 1 against do not mean a 3/1 chance.
Why quoted betting numbers need extra care
In everyday betting and gaming, quoted odds are frequently a price, showing what is paid relative to a stake, rather than a pure statement of the true chance. A quoted price of 4 to 1 behaves like odds against, but the business offering it typically builds in a margin. This means the probabilities implied by a full set of quoted prices usually add up to more than 100%. The idea is explained under bookmaker margin, and the different ways of writing prices are covered in odds formats. The key point for conversion is that the formulas above give the implied probability of a quoted price, not necessarily the real probability of the event.
The same applies in casino games: payouts are set so that, on average, the player gets back less than they stake. That built-in difference is the house edge, and it is why a game can pay out at attractive-looking odds and still cost money over time. Lottery prizes work similarly, as seen in lottery odds, where the chance of winning is tiny even though the prize is large. Raffle odds are a friendlier example, because they depend only on the number of tickets.
Practice problems
Try these, then check against the answers below.
- Convert odds in favour of 2 to 7 into a probability.
- Convert a probability of 20% into odds in favour.
- What probability corresponds to odds against of 9 to 1?
Answers:
- 2 ÷ (2 + 7) = 2/9, about 22.2%.
- 20% is 20/100 and the rest is 80/100, so the odds are 20 to 80, which simplifies to 1 to 4.
- Odds against of 9 to 1 are odds in favour of 1 to 9, so the probability is 1 ÷ 10 = 10%.
Takeaways
Probability is a share of the whole. Odds are one part compared with the other part. To convert, add the two sides of the ratio to get the whole, or subtract the probability from 1 to get the other part. Whichever form you use, the number describes a chance, not a certainty, and understanding it will not turn a game with a built-in disadvantage into a profitable one.