One chance, several ways to write it
A single chance can be expressed in many forms, and games, guides and commentators all use different ones. Being fluent in the conversions prevents misunderstanding.
Take the chance of rolling a 3 on a fair die:
- As a fraction: 1/6
- As a decimal: about 0.1667
- As a percentage: about 16.7%
- As “1 in 6”
- As odds: 5 to 1 against, or 1 to 5 for
They all describe the same thing. The first four are probabilities. The last is odds, which is a different scale and the one that most often causes confusion.
Fractions, decimals and percentages
These three are straightforward.
- Fraction to decimal: divide the top by the bottom. 1/8 = 0.125.
- Decimal to percentage: multiply by 100. 0.125 = 12.5%.
- Percentage to decimal: divide by 100. 4% = 0.04.
Some everyday values are worth knowing: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/10 = 10%, 1/20 = 5%, 1/100 = 1%.
“1 in N” is the same as the fraction 1/N. One in 200 is 1/200 = 0.005 = 0.5%. A rule of thumb is that dividing 100 by N gives the percentage when it is easy to do so: one in 25 is 100 ÷ 25 = 4%.
When comparing two percentages, remember the difference between a change in percentage points and a relative change. Going from 2% to 3% is 1 percentage point but a 50% relative increase. That distinction, covered under percentage points, matters whenever a headline says a chance “rose by half”.
Odds against and odds for
Odds compare the number of ways something can happen to the number of ways it cannot.
- Odds for an event = (chances it happens) : (chances it does not)
- Odds against = (chances it does not) : (chances it does)
For the die showing a 3: it happens in 1 way and fails in 5, so the odds for are 1 to 5 and the odds against are 5 to 1.
Odds are not the same as probability. “5 to 1 against” is not 5, and not 1/5 as a probability. Convert as follows.
From probability to odds. If the probability is p, the odds for are p : (1 − p). For p = 1/4, that is 1/4 : 3/4, which simplifies to 1 : 3. So the odds for are 1 to 3 and the odds against are 3 to 1.
From odds against to probability. For odds of a to b against, the probability is b ÷ (a + b).
- 3 to 1 against: 1 ÷ (3 + 1) = 1/4 = 25%.
- 4 to 1 against: 1 ÷ 5 = 1/5 = 20%.
- 5 to 1 against: 1 ÷ 6 ≈ 16.7%.
- 9 to 1 against: 1 ÷ 10 = 10%.
From odds for to probability. For odds of a to b for, the probability is a ÷ (a + b). Odds of 2 to 3 for give 2 ÷ 5 = 40%.
A handy check: the two numbers in the odds sum to the total number of equally likely cases. In “3 to 1 against” there are 3 + 1 = 4 cases.
Wheel examples
A European-style roulette wheel has 37 pockets (0 to 36). Taking one specific number:
- Probability: 1/37, about 2.70% (0.027027…).
- Odds against: 36 to 1.
An American-style wheel has 38 pockets because of an extra zero:
- Probability: 1/38, about 2.63%.
- Odds against: 37 to 1.
A single-number bet commonly pays 35 to 1. The gap between the true odds against (36 to 1 or 37 to 1) and the payout (35 to 1) is where the house edge comes from. Payout odds below the true odds mean the player is paid less than fair value on average. This difference in how the numbers are quoted is why it is useful to translate everything into probabilities before comparing.
Playing cards
Drawing one specific card from a 52-card deck: 1/52, about 1.92% (1 ÷ 52 = 0.01923). The odds against are 51 to 1.
Drawing any ace: 4/52 = 1/13, about 7.7%. The odds against are 48 to 4, which simplify to 12 to 1. Check: 12 + 1 = 13 cases, and 1/13 matches.
Drawing a heart: 13/52 = 1/4 = 25%. Odds against are 39 to 13, which is 3 to 1.
Reading very small chances
For tiny probabilities, “1 in N” is usually clearer than a percentage. One in 1,000 is 0.1%, while one in 14 million is about 0.0000071%, which is hard to picture. Lottery odds and other very low-probability events are better grasped by comparison, such as picturing tiny odds as a crowd of people or a row of seats.
Typical mistakes
- Treating odds as probability. “2 to 1 against” is a 1/3 chance, not 50%.
- Mixing up for and against. Always check which side the first number refers to.
- Adding percentages for repeated tries. Ten tries at 10% do not give 100%. The correct figure is 1 − 0.9^10, about 65%. This is a version of adding probabilities wrongly.
- Rounding too early. Keep fractions as long as practical, convert at the end.
A quick conversion table
| Probability | Decimal | Odds for | Odds against |
|---|---|---|---|
| 1/2 | 0.5 | 1 to 1 | 1 to 1 |
| 1/3 | 0.333 | 1 to 2 | 2 to 1 |
| 1/4 | 0.25 | 1 to 3 | 3 to 1 |
| 1/5 | 0.2 | 1 to 4 | 4 to 1 |
| 1/10 | 0.1 | 1 to 9 | 9 to 1 |
Fluent conversion lets you compare any two statements of chance fairly. It does not alter the fact that casino games carry a house edge and that outcomes are uncertain. Play only with money you can afford to lose, and seek local support if gambling becomes a problem.