Odds Calculator

probability p = for / (for + against) · odds ratio = for : against · decimal odds = 1 / p · American odds = −100p / (1 − p) when p ≥ 0.5, else +100(1 − p) / p

probability

How likely the event is, on a scale from 0 to 100 percent. This is a percentage field, so a probability of one in five is entered as 20 rather than 0.2

for

How many ways the event can happen. This is the first number of the ratio and of an odds line — the side the notation is named after, which is also the side most people get backwards

against

How many ways it can fail to happen. The two counts only matter through their ratio, so 4 : 2 and 2 : 1 say the same thing and the panel reduces them

decimal odds

The total return per one unit staked, stake included. A decimal price of 5.00 means a winning 1 comes back as 5 — the stake plus a profit of 4 — which is why it equals 1/p and not p/(1 − p)

American odds

The same price in the American convention: a positive number is the profit on a 100 stake, a negative number is the stake needed to win 100. The sign carries real information, so +400 and −400 are opposite ends of the scale, not the same number

Use it whenever a probability and a price need to become each other. Checking whether a quoted price is worth taking means turning it into a probability first — the implied probability — because a price is a claim about how likely something is, and it is hard to argue with a claim that is still in price form. Going the other way, anyone who has estimated a probability — from data, from a model, from a base rate — needs the price that estimate corresponds to before it can be compared with what is on offer. The page also settles the notation question on its own: the same 20% is 1 : 4, 5.00 and +400, and seeing all three together is the fastest way to stop misreading one of them. It is not a calculator for what to stake, and it takes no view on whether a price is good value.

1. A 20% chance written three ways

   1. Start from a 20% probability, which is 20/100 = 0.2 as a proportion
   2. The counts are 20 against 80, and both divide by 20, so the ratio is 1 : 4
   3. Decimal odds are the return per unit staked: 1 / 0.2 = 5.00, meaning a stake of 1 comes back as 5
   4. The probability is below 50%, so the American price is positive and equals 100 × (1 − 0.2) / 0.2 = 400

   Read the ratio and the price as two different things being said about the same 20%. The ratio says one of the five equally likely outcomes is the one you want; the price of 5.00 says a winning unit comes back fivefold. A bettor who takes the 1 : 4 as a price would be reading it as the bookmaker's 4/1, which is the same 20% written the other way round — and that trap is the reason both fields on this page carry their direction in the label.

2. An odds line of 4 : 2 turned into a probability

   1. The line is 4 ways for and 2 against, giving 4 + 2 = 6 equally likely outcomes in total
   2. The probability is 4 / 6 = 0.6667, or 66.67%
   3. 4 and 2 share a factor of 2, so the printed ratio reduces to 2 : 1
   4. Above 50%, the American price is negative: −100 × 0.6667 / 0.3333 = −200, meaning 200 must be staked to win 100

   The ratio here comes back reduced, which is the one place this page changes what was typed in. 4 : 2 and 2 : 1 describe the same six outcomes, and a reduced ratio is what an odds line is normally printed as. The probability and the decimal odds are unaffected: 1.5 is below 2.00 because the event is more likely than not, and a decimal price below 2.00 is exactly what odds-on means.

3. Why 65.22% comes back as 3261 : 1739

   1. 65.22% means 65.22 for every 100, and 100 − 65.22 = 34.78 against
   2. Multiply both by 100 to clear the two decimal places: 6522 and 3478
   3. Both divide by 2, and what is left — 3261 and 1739 — shares no further factor
   4. Check it: 3261 + 1739 = 5000, and 3261 / 5000 = 0.6522 exactly

   This is the honest answer and it is not a pretty one. A ratio of 3261 : 1739 is exactly the number that was typed in; the neat-looking 15 : 8 that comes close to it is a different number, and printing it would mean answering a question nobody asked. The check in the last step is the whole argument — the two counts add back up to the 5000 that 65.22% came from. If you want a round ratio, start from a round ratio: enter the odds and read the probability off instead of the other way round.

4. Odds of 1 : 100 — the same corner from the other side

   1. One way for, a hundred against, so 101 equally likely outcomes in total
   2. The probability is 1 / 101 = 0.009901, or 0.99%
   3. Decimal odds are 1 / 0.009901 = 101, so a unit staked returns 101
   4. Well below 50%, so the American price is positive: 100 × 0.990099 / 0.009901 = 10000, printed as +10000

   A longshot reverses the shape of the first example rather than changing it: the ratio stays small and readable, while the price grows to 101 and the American figure runs to five digits. The +10000 is the part worth noticing — a positive American price counts the profit on a 100 stake, so +10000 is a hundredfold profit, which is a different statement from a decimal price of 101 that includes the stake in the return.

A probability of exactly 0% or exactly 100% has no finite odds, and this page refuses to answer rather than printing an infinity. That is a property of the notation, not a gap in the calculator: odds are a ratio of two counts, and "it never happens" has no count in the numerator, while at 100% the decimal price 1/1 is fine but the American price −100p/(1 − p) divides by zero. So 0 : 4, 4 : 0 and entering 0 or 100 in the probability field are all four rejected with a message that says why. Two further limits are worth stating. The page converts a probability you supply; it does not estimate one, so a price computed here is only as good as the probability typed in, and a decimal price is not a prediction that the event has that chance — a bookmaker's price also carries a margin, so a set of prices from one book will imply probabilities that add up to more than 100%. And the reference table below the calculator is a fixed ruler of common odds, not a computation of your inputs; its title says so, and the panel is the part that follows what you typed.

A bookmaker quoted me 4/1. Why does this page say 4 : 1 is 80%?

Because the two notations put the same pair of numbers the other way round, and both are in common use. A bookmaker's 4/1 is a price: 1 staked returns 4 in profit, which corresponds to a 20% chance. Written as a ratio of counts, that same 20% is 1 : 4 — one way for, four against. This page always reads the first number of the ratio as the ways the event happens, so 4 : 1 here means four ways for against one against, which is 80%. If you are converting a quoted price, use the odds-to-probability direction and enter 1 and 4 rather than 4 and 1, or check the answer against the probability column in the table below, where the two notations are laid out side by side.

What is the difference between the ratio, the decimal odds and the American price?

They are three notations for one probability, and they answer slightly different questions. The ratio is a count — how many ways it happens against how many ways it does not — and it is the form a statistician pictures. The decimal odds is a return: 5.00 means a stake of 1 comes back as 5, stake included, so it equals 1/p. The American price switches convention at even odds: above 50% it is negative and states the stake needed to win 100, below 50% it is positive and states the profit on a stake of 100. All three move together, and the panel prints them together so that a number recognised in one notation can be checked in the other two.

Why is a 100% or 0% probability not accepted?

Because odds and prices have no finite value at the two ends. Odds are a ratio of two counts, so a certain event would need a count of zero against it and an impossible one a count of zero for it, and both of those leave the ratio undefined. The decimal price survives at 0% but not at 100%, and the American convention divides by zero at 100% while being unbounded at 0%. Rather than print an infinity, or a very large number that is not the answer, the page declines and says which of the two ends was hit. If you need to work with a certain or impossible event, the probability itself is the only finite way to write it.

Why does 65.22% come back as 3261 : 1739 rather than a rounder ratio?

Because 3261 : 1739 is the ratio that 65.22% actually is, and a rounder one would be a different number. Two counts of 3261 and 1739 add up to 5000, and 3261/5000 is exactly 0.6522 — nothing was approximated. A ratio like 15 : 8 is close to 65.22% but is not 65.22%, and it is not a line anyone quoted either, so printing it would replace the number that was typed in with one that merely resembles it. Round ratios come from round probabilities: if a neat 1 : 4 or 3 : 1 is what you want, enter the counts and read the probability off, which is the direction where the ratio is the input.

Is the decimal odds the same as the return I would get?

It is the total return per unit staked, with the stake included, so a decimal price of 1.25 at an 80% chance returns 1.25 for every 1 staked — a profit of 0.25. It is not the profit on its own, which is the decimal price minus 1, and it is not a net-of-fees figure either, so it will not match a payout that has a commission taken off it. The reason to use the stake-inclusive form is that it is the one printed on betting sites, and reading it as pure profit is the commonest way the number gets misread: a price of 1.25 looks like a 125% return and is actually a 25% profit.

Does the table under the calculator change when I change the inputs?

No, and it is not meant to. The table is a fixed ruler for the odds lines that come up most often, so it stays put while the panel above it follows whatever was typed in. That is why the two can disagree without either being wrong — the panel answers your inputs, and the table answers the general question of what the familiar lines are worth. Its title says as much, and the direction it uses is the same as the panel's, with the ways it happens first and the ways it does not second, so the two can be read side by side without switching conventions.

• Odds — from Wolfram MathWorld (betting odds written as a ratio a to b, and the conversion between that ratio and the probability a/(a + b)) — Wolfram MathWorld
• Interpretations of Probability — Stanford Encyclopedia of Philosophy (section 3.3.2, the betting analysis of degrees of belief: credence identified with the betting odds a person regards as fair, which is the step this page's odds-to-probability direction formalises) — Stanford Encyclopedia of Philosophy
• 1.3.6.1. What is a Probability Distribution — e-Handbook of Statistical Methods (the frequency reading of probability that the for/against counts here are a form of) — National Institute of Standards and Technology (NIST)