Probability and odds describe the same chance in two different languages. This guide moves between them step by step, then uses roulette and a lottery to show what the numbers cost.
It is part of our Odds, probability and strategy section.
What is the difference between probability and odds?
Probability is the number of ways an event can happen divided by the number of possible outcomes, a value from 0 to 1. Odds are the ratio of the ways it happens to the ways it does not. A fair coin has a probability of 1/2 for heads and odds of 1 to 1.
The two are often mixed up. A 2014 piece from Sense About Science USA points to a news story that called a 74% chance “odds”. Its fix is a 74% probability or odds of about 3 to 1.
Betting adds a third meaning: “odds” often means the payout, as in “pays 35 to 1”. Mathwords stresses that this describes the payout, not the chance of winning. A payout can sit below the true odds, and that gap is the House edge The share of each bet a casino expects to keep on average over the long run. It is set by the rules of the game and does not depend on luck. Read the full entry. Here, “odds against” means the chance ratio and “pays” means the payout.
How do you calculate a probability?
Divide the number of ways the event can happen by the total number of equally likely outcomes. One face of a die shows a 4 out of six faces, so the probability is 1/6, or 16.67%. Multiply by 100 for a percentage, as Maths Is Fun explains.
The table shows six common cases.
| Event | Ways / outcomes | Percentage |
|---|---|---|
| Coin lands heads | 1/2 | 50.00% |
| Die shows a 4 | 1/6 | 16.67% |
| Two dice show a double | 6/36 | 16.67% |
| Card drawn is a club | 13/52 | 25.00% |
| Roulette: one number, single zero | 1/37 | 2.70% |
| Roulette: red, single zero | 18/37 | 48.65% |
How do you turn odds into a probability?
For odds against of X to Y, add the two numbers and divide Y by the total: probability = Y / (X + Y). Odds of 5 to 1 against give 1/6, or 16.67%. To go the other way, odds against equal (1 minus p) divided by p, where p is the probability.
- Write the odds against as X to Y, such as 36 to 1.
- Add the numbers: 36 + 1 = 37.
- Divide the second number by the total: 1 / 37 = 0.0270, or 2.70%.
See the Wikipedia article on odds. Odds in favor flip it: 1 to 5 in favor is 5 to 1 against.
A common slip is dividing by the wrong number. Odds of 5 to 1 against are 1/6, not 1/5, because both parts go into the total.
How do decimal, fractional and American odds fit together?
Fractional, decimal and American odds are three ways to write the same price. Decimal odds Odds written as a single number showing the total return, stake included, for each 1 unit staked. Common outside the US. Read the full entry include your stake, so a fair decimal price is 1 divided by the probability. Decimal equals fractional plus 1. American odds show the profit on a 100 stake (plus) or the stake needed to win 100 (minus).
The table shows the fair price in each format, meaning what a bet would pay with no house edge.
| Event | Probability | Percentage | Fractional (against) | Decimal | American |
|---|---|---|---|---|---|
| Coin lands heads | 1/2 | 50.00% | 1/1 | 2.00 | +100 |
| Card is a club | 13/52 | 25.00% | 3/1 | 4.00 | +300 |
| Die shows a 4 | 1/6 | 16.67% | 5/1 | 6.00 | +500 |
| Roulette: red, single zero | 18/37 | 48.65% | 19/18 | 2.06 | +106 |
| Roulette: one number, single zero | 1/37 | 2.70% | 36/1 | 37.00 | +3600 |
To read a price the other way, divide 1 by the decimal figure. A price of 1.65 implies 1 / 1.65 = 60.6%. This is the Implied probability The chance of an outcome that is built into its odds. In decimal odds it is 1 divided by the odds, shown as a percentage. Read the full entry.
Real prices are shorter than fair ones. In Wikipedia’s example, prices imply 60%, 50% and 20% for three outcomes. They add to 130%, a margin called the overround.
How do you combine probabilities?
For independent events, multiply the probabilities. Three heads in a row is 1/2 × 1/2 × 1/2 = 0.125, or 12.5%. For “at least one”, work out the chance of none and subtract it from 1. Both need independent events.
Two worked cases show “at least one” (Maths Is Fun covers the multiplication rule).
- At least one 6 in four rolls of a die. The chance of no 6 on one roll is 5/6. For four rolls: (5/6)4 = 0.4823. So the chance of at least one 6 is 1 – 0.4823 = 0.5177, or 51.77%.
- A given roulette number at least once in 37 spins. The chance of a miss on one spin is 36/37. For 37 spins: (36/37)37 = 0.3629. So the chance of at least one hit is 1 – 0.3629 = 63.71%.
Thirty-seven spins is one per pocket on average, yet a given number shows up only about 64% of the time.
Why do earlier results not change the next spin?
Roulette spins, coin tosses and dice rolls are Independent events Events where the outcome of one has no effect on the chance of the other, such as two separate roulette spins or two coin flips. Read the full entry: each has no memory of the one before. On a single-zero wheel, the chance of red is 18/37, or 48.65%, after any run of reds or blacks. Expecting the opposite is the Gambler’s fallacy The mistaken belief that a random result is due to change because of what came before, such as black being due after several reds. Read the full entry.
As Wikipedia recounts, on August 18, 1913, at the Monte Carlo Casino, the ball fell on black 26 times in a row. Gamblers reportedly lost millions of francs betting that red was due.
A run of one color 26 times is 2 × (18/37)26, about 1 in 68.4 million (we get 1 in 68,411,592). That is the chance before the run starts. After 25 blacks, the next spin is still black with probability 18/37.
How does a payout compare with the true odds?
Compare what a bet pays with the true odds against it. On a single-zero wheel a straight-up number pays 35 to 1, but the true odds against are 36 to 1. The shortfall of one unit per 37 spins on average is the house edge: 1/37, or 2.70%. Every standard bet has the same gap.
Payouts come from New Jersey’s roulette payout rule and the Nevada Gaming Control Board rules of play. Suppose 37 spins land on every pocket once. A straight-up bet wins 35 units once and loses 1 unit 36 times: 35 – 36 = -1 unit, or -1/37 = -2.70% per spin.
| Bet | Numbers | True odds against (single zero) | Pays | Numbers × (pays + 1) | Edge, single zero | Edge, double zero |
|---|---|---|---|---|---|---|
| Straight up (one number) | 1 | 36 to 1 | 35 to 1 | 36 | 2.70% | 5.26% |
| Split | 2 | 35 to 2 | 17 to 1 | 36 | 2.70% | 5.26% |
| Street | 3 | 34 to 3 | 11 to 1 | 36 | 2.70% | 5.26% |
| Corner | 4 | 33 to 4 | 8 to 1 | 36 | 2.70% | 5.26% |
| Six line | 6 | 31 to 6 | 5 to 1 | 36 | 2.70% | 5.26% |
| Dozen or column | 12 | 25 to 12 | 2 to 1 | 36 | 2.70% | 5.26% |
| Even-money bet, such as red | 18 | 19 to 18 | 1 to 1 | 36 | 2.70% | 5.26% |
Look at the fifth column. Every bet gives numbers × (pays + 1) = 36, but the wheel has 37 pockets. That missing pocket is the whole edge. With 38 pockets against the same 36, a double-zero wheel has an edge of 2/38 = 5.26%. Our roulette odds and payouts guide has more.
What does the house edge cost over time?
The House edge The share of each bet a casino expects to keep on average over the long run. It is set by the rules of the game and does not depend on luck. Read the full entry becomes a cost per hour once you add a bet size and a pace. At 40 spins an hour and $10 a spin, you stake $400 an hour. At 2.70% the average cost is about $10.81 an hour on a single-zero wheel, and about $21.05 on a double-zero wheel.
The pace of 40 spins an hour is an illustrative assumption, not a measured figure.
| Wheel and bet | House edge | Average cost at $10 a spin, 40 spins an hour |
|---|---|---|
| Single zero, any standard bet | 2.70% | $10.81 |
| Double zero, any standard bet | 5.26% | $21.05 |
An average hides swings. How likely is a player to be ahead after a set number of spins on one straight-up number, at 1 unit a spin?
| Spins | Hits needed to be ahead | Average hits | Chance of being ahead | Average loss (units) |
|---|---|---|---|---|
| 37 | 2 | 1.0 | 26.4% | 1.0 |
| 100 | 3 | 2.7 | 50.9% | 2.7 |
| 370 | 11 | 10.0 | 41.7% | 10.0 |
| 1,000 | 28 | 27.0 | 45.1% | 27.0 |
After 37 spins, about 26.4% of players are ahead, since two hits are needed and the average is one. After 100 spins it is 50.9%, because three hits are enough. The average loss keeps growing.
Being ahead after a short session is variance, not a lower edge. The Expected value The average amount a bet is expected to win or lose per bet over a very large number of repeats. Casino games have a negative expected value for players. Read the full entry of every spin stays at -1/37 of the stake.
How do probability and odds look in a lottery?
Lottery odds are usually quoted as 1 in N, which is a probability of 1/N. Powerball’s jackpot odds are 1 in 292,201,338 for a $2 play, a probability of about 0.00000034%. As odds against, that is 292,201,337 to 1. See the Powerball prize chart.
The number comes from counting. The format draws five white balls from 69 and one red ball from 26, per the Powerball FAQs. There are 11,238,513 ways to pick five of 69, and 11,238,513 × 26 = 292,201,338.
The same chart gives overall odds of winning any prize as 1 in 24.87, which is 4.02%. Buying 100 plays raises the jackpot chance about 100-fold, to roughly 1 in 2.9 million. Each draw is independent.
Which mistakes trip people up most?
The usual slips are treating 5 to 1 as 1/5, confusing profit with total return, reading a payout as the true odds, and reading a long-run average as a forecast for the next spin. Each has a quick check, listed below with the numbers.
- Odds read as probability. 5 to 1 against is 1/6 (16.67%), not 1/5 (20%). Add both numbers before dividing.
- Profit mistaken for total return. At 3 to 1 on a $10 stake, the profit is $30 and the total returned is $40, as Mathwords points out.
- Payout mistaken for odds. Paying 35 to 1 on a 36 to 1 shot is a 2.70% cost, not a fair price.
- An average read as a forecast. A 2.70% average cost says nothing about the next spin.
What should you try next?
Pick one bet you have seen priced and run these checks before you play. Convert the price, compare it with the true probability, work out the hourly cost and set a limit first. Take a few minutes with a calculator and these five steps.
- Convert the price: divide 1 by a decimal price, or use Y / (X + Y) for X to Y against.
- Work out the true probability from the rules, such as 18/37 for red on a single-zero wheel.
- If the implied probability is higher than the true one, the bet costs you on average.
- Multiply your hourly stake by the edge for the average hourly cost.
- Choose a spending and time limit before you start.
If you want help setting limits, the Safer play toolkit explains how. Our guide to the house edge covers the full cost of a bet.