Why conversion matters
Casino games are games of chance with a built-in house edge. No strategy, system, pattern or timing changes the odds in the player’s favor over time, and most players lose money over time. Nobody should gamble money they cannot afford to lose.
Chances are described in several ways: as a fraction (1/6), a percentage (16.7%), a ratio (5 to 1 against), or decimal odds. Being able to translate between them helps you read what a number actually says, rather than being swayed by how it is written. The examples below are hypothetical or textbook cases, worked by hand.
Probability as a fraction, decimal and percentage
These three are the easy conversions.
- 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 fraction: put it over 100 and simplify. 25% = 25/100 = 1/4.
A single-zero roulette number is 1/37, which is 1 ÷ 37 ≈ 0.0270, or about 2.70%. Rounding is acceptable for reading, but be careful when multiplying rounded figures, because small errors grow.
From odds against to probability
Odds are a different thing from probability. Odds compare the unfavorable outcomes with the favorable ones. Probability compares the favorable ones with all of them.
For odds of “a to b against” (a unfavorable for every b favorable):
probability = b ÷ (a + b)
Examples:
- 3 to 1 against: probability = 1 ÷ (3 + 1) = 1/4 = 25%.
- 5 to 2 against: probability = 2 ÷ (5 + 2) = 2/7 ≈ 28.57%. Check: 2 ÷ 7 = 0.2857.
- 1 to 1 (often called “evens”): 1 ÷ 2 = 50%.
- 36 to 1 against: 1 ÷ 37 ≈ 2.70%. This matches the single roulette number.
If the odds are given in favor, as “a to b on”, then the roles swap: probability = a ÷ (a + b). So 4 to 1 on (four favorable for each unfavorable) gives 4 ÷ 5 = 80%. In some betting contexts, such odds are written as 1/4, with the short side first. Always check whether a figure is quoted for or against.
From probability to odds against
To go the other way, divide the chance of not happening by the chance of happening:
odds against = (1 − p) ÷ p
Examples:
- p = 20%: (0.8) ÷ (0.2) = 4, so 4 to 1 against.
- p = 1/6 (rolling a particular number on a die): (5/6) ÷ (1/6) = 5, so 5 to 1 against.
- p = 12.5% = 1/8: (7/8) ÷ (1/8) = 7, so 7 to 1 against.
- p = 40%: (0.6) ÷ (0.4) = 1.5, which is 3 to 2 against.
A shortcut for a fraction p = 1/n is that the odds against are (n − 1) to 1. For p = 1/37, that is 36 to 1.
Decimal odds
Decimal odds, common in some regions, state the total return per unit staked, including the stake. To convert fractional odds a/b to decimal, compute a/b + 1.
- 5/2 becomes 2.5 + 1 = 3.5.
- 3/1 becomes 4.0.
To find the implied probability from decimal odds, take 1 ÷ decimal. For 3.5, that is 1 ÷ 3.5 ≈ 28.57%, matching the earlier 5 to 2 example. For 4.0, it is 25%.
Odds “to” versus “for”
Some payouts are written “35 to 1” and others “36 for 1”. These mean the same thing. With “35 to 1”, a $1 winning bet earns $35 in profit and keeps the stake, for $36 total. “36 for 1” counts the returned stake in the figure. Confusing the two can make a payout seem more or less generous than it is.
Payout odds versus true odds
The conversion above tells you the true odds implied by a probability. The payout offered on a bet is a separate number. For a single-zero roulette number, the true odds against are 36 to 1, but the payout is 35 to 1. Putting both into percentages makes the difference visible: true odds correspond to 1/37, or 2.70%, while a payout of 35 to 1 would be fair only if the chance were 1/36, or 2.78%. That gap is a source of the house edge, and the same comparison works for payout ratios on any bet.
A hypothetical check: a bet is offered at 4 to 1 on an event that has a 1 in 6 chance. The implied probability of the payout is 1 ÷ 5 = 20%, while the true probability is about 16.7%. The payout is set as if the event were more likely than it is, which makes it look reasonable at a glance. The expected value of a $1 bet is (1/6 × 4) − (5/6 × 1) = 4/6 − 5/6 = −1/6, about −$0.17.
Combining independent chances
To find the probability of several independent events all occurring, multiply the probabilities, not the odds. Two independent 25% events both happening is 0.25 × 0.25 = 0.0625, or 6.25%. In odds form, that is 15 to 1 against (since 0.9375 ÷ 0.0625 = 15), not 3 to 1 times 3 to 1.
Common mistakes
- Using a ÷ b rather than b ÷ (a + b) when converting odds against. 3 to 1 against is 25%, not 33%.
- Mixing odds for and against.
- Adding percentages for events that are not mutually exclusive.
- Assuming a payout ratio shows the real chance. It does not; it is set by the operator.
- Treating a rounded percentage as exact in a later calculation.
Conversion also helps with very small chances. Lottery odds quoted as “1 in 10 million” and a roulette hit of 1 in 37 sound like different kinds of things, but converting both to percentages (0.00001% and about 2.70%) puts them on one scale. The way a chance is presented can shift how it feels, and framing effects are a well-known reason to convert before judging an offer.
Learning to translate between these forms is mostly a matter of practice. Once it is routine, the numbers attached to a game become easier to question, and the expected loss built into every bet is easier to see.