Odds vs Probability: What's the Difference and How to Convert Between Them

Odds vs probability comes down to what you compare. Probability compares the ways you can win with all possible outcomes. Odds compare the ways you can lose with the ways you can win. A single number on a European roulette wheel has a probability of 1 in 37 (2.70%), but odds of 36 to 1 against. Same chance, two different ways of writing it.

The difference matters at the casino because games quote payouts as odds. Compare the payout odds with the true odds and you can work out the house edge of any bet yourself. This guide gives you the formulas and works through real examples with roulette and dice numbers we calculated.

What probability means

Britannica's entry on gambling defines probability as the number of favourable outcomes divided by the total number of possible outcomes, when every outcome is equally likely:

Probability always sits between 0 (impossible) and 1 (certain), often written as a percentage. OpenStax's statistics textbook adds the long-run view: probability is the relative frequency you would see over a very large number of repetitions. A 2.70% chance means that, over many thousands of spins, a given number comes up about 2.70% of the time.

What odds mean

Britannica defines odds as the ratio of unfavourable outcomes to favourable outcomes. These are odds against:

You'll also see odds in favour, which is the same ratio flipped (1 to 36 in favour). Casinos and bookmakers almost always quote odds against, because that's how payouts are written: "pays 35 to 1" means you win 35 units for each unit staked, and get your stake back.

The difference between odds and probability, side by side

The most common mix-up is treating "1 in 5" and "5 to 1" as the same. They aren't: 1 in 5 is a 20% probability, while 5 to 1 against is 1 ÷ 6 = 16.67%.

Conversion formulas: odds and probability

These four formulas cover almost every conversion you'll need:

The last one is the key to casino math. "Payout odds + 1" is your total return per unit staked, so p × (payout + 1) is the average return per unit. Whatever falls short of 1 is the house edge.

Worked examples with roulette

Roulette is the cleanest example because every pocket is equally likely. Here is every European-wheel bet from our roulette dataset, with the true odds computed from the number of pockets covered:

Take the dozen bet: 12 numbers win and 25 lose, so the true odds are 25 to 12 against, and the probability is 12 ÷ 37 = 32.43%. It pays 2 to 1. Using the formula: 1 − (12/37) × 3 = 1 − 36/37 = 2.70%. Every standard bet gives the same answer, because each pays as though the zero weren't there.

Now the American wheel, with 38 pockets. A single number still pays 35 to 1, but the true odds become 37 to 1:

The Wizard of Odds lists the same edges: 2.70% for single zero and 5.26% for double zero. Their roulette page also flags the American five-number bet (0, 00, 1, 2, 3) as worse, at 7.89%. Our data agrees: probability 13.158%, pays 6 to 1, house edge 7.89%. You can see the full comparison in our roulette risk and reward guide.

Worked examples with dice

Two six-sided dice. Each die has 6 faces, so there are 6 × 6 = 36 equally likely combinations. OpenStax's multiplication rule says that for independent events, P(A and B) = P(A) × P(B), so any one specific combination has probability 1/6 × 1/6 = 1/36.

A 7 can be made 6 ways, so its probability is 6/36 = 16.67% and its odds are 30 to 6 against, which simplifies to 5 to 1. Our million simulated rolls landed on 7 in 16.62% of cases.

Crypto dice. Crypto dice games skip the ratio and quote a multiplier: your total return per unit staked, also called decimal odds. Our dice data uses a 1% house edge, so the multiplier is 99 ÷ win chance:

At a 49.5% win chance, a fair game would pay 2.0202x. The game pays 2.00x, and 1 − 0.495 × 2 = 0.01, or 1%. You can test any other win chance with our dice odds calculator, and our dice game guide covers the bets in more detail.

Payout odds vs true odds: where the house edge lives

Every example above follows the same pattern:

1. Work out the probability from the outcomes.
2. Convert it to true odds (what a fair game would pay).
3. Compare with the payout odds the game actually offers.
4. The shortfall, weighted by the probability of winning, is the house edge.

That's why no betting pattern can fix a bad bet. The edge is built into each payout. Our roulette strategy simulations show that systems like Martingale lose the same share of money wagered as flat betting, and the baccarat strategy guide applies the same comparison to the banker, player and tie bets.

Common mistakes

• Reading "1 in 5" as "5 to 1". The first is 20%; the second is 16.67%.
• Forgetting the stake in a multiplier. A 2x multiplier is a 1 to 1 payout, not 2 to 1.
• Assuming payout odds are true odds. In a casino they're set slightly below.
• Thinking past results change the probability. Independent spins and rolls don't remember each other.

For more on how the edge plays out across every game we've calculated, see the odds and house edge hub.