Odds Calculator - Odds to Probability & Back

About the Odds Calculator

Odds and probability describe the same thing — how likely something is — in two different ways. Probability compares successes with all possible outcomes: a 25 percent chance is 1 in 4. Odds compare successes with failures: the same chance is 1 to 3 in favour, or 3 to 1 against. Bookmakers, gamblers and some statisticians work in odds; most other people think in percentages, so converting between them is a common need.

This odds calculator converts odds for or against into probability, decimal odds into implied probability, and probability into odds. For each, it shows the fractional, decimal and American formats used by bookmakers around the world, and the fair payout on a stake of 10 — what a bet would pay if the odds exactly matched the probability.

How to Use the Odds Calculator

Choose Odds to probability, Decimal odds or Probability to odds.

For odds, enter the two numbers — for 3 to 1, enter 3 and 1 — and say whether the odds are against (the bookmaker style) or in favour.

For decimal odds, enter the number, such as 2.50.

For probability, enter the percentage chance.

The Formulas

  odds a to b against   →  P = b ÷ (a + b)
  odds a to b for       →  P = a ÷ (a + b)
  decimal odds d        →  P = 1 ÷ d
  probability P         →  odds for = P : (1 − P)

  American odds:  +100 × (d − 1) when d ≥ 2
                  −100 ÷ (d − 1)  when d < 2

Step-by-Step Example: Odds to Probability

Odds of 3 to 1 against.

  3 ways to lose, 1 way to win, 4 in total
  P = 1 ÷ (3 + 1) = 0.25 → 25%
  Decimal 4.00 · American +300

A fair bet of 10 at 3 to 1 would win 30 profit, returning 40 including the stake.

Step-by-Step Example: Probability to Odds

A 20% chance.

  20 : 80 = 1 : 4 in favour
  → 4 to 1 against, fractional 4/1, decimal 5.00, American +400

A 60% chance.

  60 : 40 = 3 : 2 in favour
  → 3 to 2 on (2/3 against), decimal 1.67, American −150

When an outcome is more likely than not, bookmakers call the odds "on" rather than "against".

Step-by-Step Example: Decimal Odds

Decimal odds of 2.50.

  P = 1 ÷ 2.5 = 0.4 → 40%
  Fractional: 1.5 to 1 = 3/2 · American +150

The Three Betting Formats

Fractional odds, traditional in Britain and Ireland, show profit relative to stake: 3/1 wins 3 for every 1 staked. Decimal odds, common in Europe, Australia and Canada, show the total return per unit staked, including the stake: 4.00 returns 4 for every 1. American odds use 100 as a base: a plus figure is the profit on a 100 stake, and a minus figure is the stake needed to win 100. All three describe exactly the same price, so the calculator shows them together.

Odds in Statistics

Outside betting, odds are important in medicine and research. The odds ratio compares the odds of an outcome between two groups — for example, the odds of disease among smokers and non-smokers — and is the natural result of logistic regression and case-control studies. Odds and probability are close when events are rare: a 1 percent probability is odds of 1 to 99, which is about 0.0101. For common events they diverge sharply: a 50 percent probability is odds of 1, and a 90 percent probability is odds of 9.

The Bookmaker's Margin

Real bookmakers' odds are not fair odds. If a tennis match is a genuine 50–50, fair odds would be 2.00 for each player, but a bookmaker might offer 1.91 on both. The implied probabilities are then 52.36 percent each, adding up to 104.7 percent. The extra 4.7 percent is the bookmaker's margin, also called the overround or vig. To estimate the true probabilities, divide each implied probability by the total.

Converting Fractional Odds Quickly

A handy shortcut: to turn fractional odds against into a probability, divide the second number by the sum of both. So 5/1 is 1 ÷ 6, about 16.67 percent; 7/2 is 2 ÷ 9, about 22.22 percent; and evens, 1/1, is 1 ÷ 2, exactly 50 percent. Odds-on prices work the same way: 1/2 is 2 ÷ 3, about 66.67 percent. To go the other way, add one to fractional odds to get decimal odds — 5/1 becomes 6.00 and 1/2 becomes 1.50 — and subtract one from decimal odds to get the fractional profit per unit staked.

Expected Value of a Bet

Converting odds into probability also shows whether a bet is worth making. The expected value is the probability of winning times the profit, minus the probability of losing times the stake. If you believe a horse has a 25 percent chance, fair odds are 3/1. Offered 4/1 on a stake of 10, the expected value is 0.25 × 40 − 0.75 × 10 = 2.50, a positive bet; offered 2/1, it is 0.25 × 20 − 0.75 × 10 = −2.50, a losing one in the long run.

Understanding Your Result

The headline is the probability, or the odds when you start from a probability.

The probability line gives the chances of the event happening and not happening.

The odds line gives the odds against and in favour.

The formats line gives the fractional, decimal and American equivalents.

The fair payout line shows what a stake of 10 would return at fair odds.

When Should You Use This Calculator?

Use it to understand what a bookmaker's price implies about the chance of winning.

Use it to convert between fractional, decimal and American odds.

Use it to express a probability as odds for a report or study.

Use it for homework on odds and probability.

Common Mistakes

Treating odds as probabilities. Odds of 1 to 4 mean a 20 percent chance, not 25.

Mixing up for and against. Bookmakers quote odds against unless they say "on".

Forgetting the stake in decimal odds. Decimal odds include it; fractional odds do not.

Ignoring the margin. Bookmakers' prices always imply a total above 100 percent.

Reading a minus American price as a loss. It shows the stake needed to win 100.