Read this first
Slot machines and other games of chance have a built-in house edge, and no strategy, system or timing changes the odds. Most players lose money over time, and nobody should gamble money they cannot afford to lose. This article is a numeracy exercise. All examples are hypothetical, and nothing here is a way to win or a suggestion to play.
Being able to move between odds and percentages is a handy skill because chances are quoted in both forms. The arithmetic is short once you know whether the odds are “for” or “against”.
Two quick definitions
Odds against of a to b (written a:b) mean a unfavorable outcomes for every b favorable ones. Odds in favor of a to b mean the reverse. The difference between odds and probability is explored on its own, so here we just need the procedure.
From odds against to a percentage
Odds against of a to b: the total number of outcomes is a + b, and b of them are favorable.
Step 1. Add the two numbers. Step 2. Divide the second number (the favorable side) by that total. Step 3. Multiply by 100.
Example: odds of 3 to 1 against.
- 3 + 1 = 4
- 1 ÷ 4 = 0.25
- 0.25 × 100 = 25%
More examples:
- 4 to 1 against: 1 ÷ 5 = 20%
- 9 to 1 against: 1 ÷ 10 = 10%
- 2 to 1 against: 1 ÷ 3 = 0.3333, which is about 33.3%
- 3 to 2 against: 2 ÷ 5 = 40%
- 5 to 2 against: 2 ÷ 7 = 0.2857, which is about 28.6%
Notice that “4 to 1 against” is not 25%. Treating the first number as the denominator is the most common error, and it gives wrong answers every time.
From odds in favor to a percentage
For odds in favor of a to b, the favorable side is the first number. So divide a by a + b.
Example: odds of 3 to 1 in favor.
- 3 + 1 = 4
- 3 ÷ 4 = 0.75
- 0.75 × 100 = 75%
“1 to 1” odds, often called even odds, give 1 ÷ 2 = 50%, the same as a fair coin.
A good sanity check is that “in favor” odds above 1 to 1 must give more than 50%, and “against” odds above 1 to 1 must give less than 50%.
From a percentage back to odds
Go the other way by comparing the percentage chance of the event with the percentage chance of it not happening.
Step 1. Subtract the percentage from 100 to find the chance that it does not happen. Step 2. Write the two as a ratio. Step 3. Simplify.
Example: a 20% chance.
- 100 − 20 = 80
- Odds in favor are 20 to 80
- Dividing both by 20 gives 1 to 4 in favor, which is 4 to 1 against.
More examples:
- 75%: 75 to 25 in favor, which simplifies to 3 to 1 in favor
- 12.5%: 12.5 to 87.5, which simplifies to 1 to 7 in favor, or 7 to 1 against
- 40%: 40 to 60, which simplifies to 2 to 3 in favor, or 3 to 2 against
“1 in N” statements
A “1 in N” statement is already a probability. Divide 1 by N for the decimal and multiply by 100 for the percentage.
- 1 in 8: 1 ÷ 8 = 0.125, so 12.5%
- 1 in 20: 5%
- 1 in 250: 0.4%
To convert “1 in N” to odds against, use N − 1 to 1. So 1 in 6 is 5 to 1 against, and 1 in 20 is 19 to 1 against.
A rounding warning
Rounded results can hide detail. 2 to 1 against is 33.3% recurring, not exactly 33%. For small chances, the gap matters more in relative terms. A chance of 0.4% rounded to “under 1%” looks like it might be almost anything. Keep a couple of decimals until the end, and note what you rounded. The same caution applies when you read a figure in a promotion. Rounding and selective presentation are among the ways advertised chances can mislead.
Checking your answers
A reliable test is to convert your answer back. If 3 to 2 against gave 40%, then 40% should return 40 to 60, or 2 to 3 in favor, which is 3 to 2 against. If it does not round-trip, something went wrong.
Another check: the probability of the event and the probability that it does not happen must add to 100%. For 4 to 1 against, 20% and 80% do.
Where different notations come in
Odds are written in several styles, such as fractions like 5/2, decimals like 3.50 and American numbers such as +250. Each one has its own conversion rule. That is a topic of its own, covered under odds formats.
Why this skill helps
Being able to convert is useful for comparing claims. It also makes the idea of expected value clearer, because average results are built from probabilities. And it can make small or large chances seem more real, which counters some of the cognitive biases that distort how we judge risk.
Remember that understanding the arithmetic does not change the result of a game. If you do gamble, treat it as paid entertainment, set limits before playing, never chase losses and take regular breaks. If gambling worries you, a national problem-gambling helpline or local health service can offer confidential help.