Calculating 1X2 and Draw Probabilities with Poisson and EV

14 November 2026

A football match that finishes level is the hardest of the three 1X2 outcomes to price, because it depends on both teams scoring the same number of goals, whatever that number is. The standard way to build that probability is the Poisson model, which treats the home and away goal counts as independent. Draw probability then comes from adding up every equal-score outcome. Once you can do that, you can compare your numbers with a bookmaker's prices and test whether a bet is worth its odds.

From expected goals to exact scorelines

The Poisson model needs just two inputs: the number of goals the home side is expected to score, and the number the away side is expected to score. Call them λ for the home team and μ for the away team. For any team, the probability of scoring exactly k goals is e^(-λ) × λ^k / k!, where e is the mathematical constant of about 2.718 and k! is k factorial. As footballdatalab.co.uk explains, the standard model treats the two teams' goal counts as independent, so the probability of a particular scoreline is simply the home figure multiplied by the away figure.

Take an illustrative match where the home side is expected to score 1.4 goals and the away side 1.1. The chance of 0-0 is e^(-1.4) × e^(-1.1), which is e^(-2.5), or about 8.2%. The chance of 1-1 is 1.4 × 1.1 × e^(-2.5), about 12.6%. A 2-2 comes out near 4.9%, a 3-3 near 0.8%, and a 4-4 under 0.1%.

The 1X2 probabilities are built by grouping those scorelines. Every scoreline where the home team has more goals goes into the home win. Every scoreline where the away team has more goes into the away win. Draw probability is the sum of all the equal scores: 0-0, 1-1, 2-2 and so on. With the figures above, that total comes to a little under 27%, with the rest split between the two sides according to their goal rates.

In practice, a spreadsheet does this easily. You build a grid of home goals against away goals, typically 0 to 6 or so each way, fill each cell with the product of the two Poisson probabilities, and sum the diagonal for the draw. Cells above and below the diagonal give you the two win probabilities. Stopping the grid at a reasonable number of goals loses only a sliver of probability, so the three totals will sit very slightly under 100% unless you scale them back up.

Why draws need a correction

Independence is a convenient assumption, but real matches rarely behave so tidily, and the trouble shows up at the lowest scores. That is where Dixon and Coles stepped in. Their football model leaves the Poisson framework in place but adjusts the four lowest scorelines: 0-0, 1-0, 0-1 and 1-1. Every other scoreline keeps its ordinary Poisson probability.

The adjustment is controlled by a parameter written ρ (rho). According to the Dixon-Coles explanation published by Statometrics, a negative ρ increases the probabilities of 0-0 and 1-1 and decreases those of 1-0 and 0-1. In other words, probability moves out of the narrow one-goal results at the bottom of the scale and into the two low-scoring draws. Because draws are made up of exactly those scorelines, plus higher ones, the correction nudges the overall draw probability upwards when ρ is negative.

For a reader, the consequence is straightforward. A plain Poisson draw figure, such as the 27% above, is a starting point rather than a finished answer. If you believe low-scoring matches in a particular league cluster around level results more than independence would suggest, the correction is the tool for expressing that belief. The size of ρ is something you estimate from data, not a number to pick by instinct, and a small value will leave your results close to plain Poisson.

The adjusted formula is compact. The probability of the home team scoring x and the away team scoring y becomes τ(x,y) × (λ^x e^(-λ) / x!) × (μ^y e^(-μ) / y!), as set out in the methodology page on impliedscore.com. The factor τ(x,y) equals 1 for every scoreline outside the four low ones, so the standard Poisson calculation is left untouched there.

Turning bookmaker odds into fair probabilities

A model gives you a view of the match. To test it, you need the bookmaker's view in the same units, and that means stripping out the margin. A bookmaker's raw implied probabilities in a 1X2 market normally add up to more than 100%, and the excess is called the overround. It is the built-in margin, and it is why backing every outcome at the quoted prices guarantees a loss.

UK fractional odds need converting first. Divide the first number by the second and add 1, so 11/10 becomes 2.10 in decimal. Using illustrative decimal prices of 2.10 for the home win, 3.40 for the draw and 3.50 for the away win, the calculation runs as follows.

• Raw implied probability is 1 divided by the decimal odds: 47.6% home, 29.4% draw, 28.6% away.
• The overround is the total minus 100%: these three add up to 105.6%, so the margin is 5.6 percentage points.
• Proportional removal divides each raw probability by that total of 1.056, so the fair figures add up to exactly 100%.
• Fair probabilities in this example are about 45.1% home, 27.9% draw and 27.1% away.
• The fair draw price is 1 divided by 0.2785, roughly 3.59, against the 3.40 actually offered.

Proportional removal, as impliedscore.com describes it, is the simplest method: it scales every outcome by the same factor. That is easy to reproduce and good enough for most purposes, though it assumes the margin is spread evenly across the outcomes. The gap between the 3.59 fair price and the 3.40 on offer is the practical cost of the margin on a draw bet.

Fitting implied expected goals from the market

With fair 1X2 probabilities in hand, you can work backwards from the market to the goal rates it implies. The worked example on impliedscore.com follows this order: it converts the 1X2 market probabilities to fair probabilities first, then fits expected goals, then calculates score probabilities. The order matters. If you fitted to raw probabilities that include the margin, the resulting goal rates would describe a match that does not exist.

The idea is a search. You pick trial values of λ and μ, feed them into the Dixon-Coles formula, sum the scorelines into home, draw and away probabilities, and compare those with the fair market figures. You then adjust λ and μ until the model's three totals sit as close as possible to the market's. A spreadsheet solver can do the job, as can a few lines of code. The pair of numbers that results is the market-implied expected goals for each side.

Those implied rates are useful in two ways. They tell you what scoring pattern the bookmaker is effectively pricing, which you can sanity-check against recent form. They also give you a full grid of scoreline probabilities consistent with the market, so you can see how much of the draw price comes from 0-0 and 1-1 rather than from higher scores. If your own view of the match differs from the implied goal rates, you know exactly where your disagreement lies.

One practical caveat: the market gives three probabilities, but two of them are independent because the three must sum to 100%. Two unknowns, λ and μ, can therefore be pinned down from them, but with ρ also free the fit is no longer unique. In practice ρ is usually fixed from historical data first and only the two goal rates are fitted.

Using expected value to test a bet

Expected value, or EV, puts a number on whether a price is worth taking. As impliedscore.com sets out, you multiply the win probability by the net return, then subtract the stake weighted by the probability of losing. For a £1 stake at decimal odds of 3.40, the net return on a win is £2.40.

Suppose your own model, after the Dixon-Coles adjustment, puts the draw at 30%. The EV is 0.30 × £2.40 minus 0.70 × £1, which is £0.72 minus £0.70, or +£0.02 per £1 staked. The break-even point at 3.40 is a probability of about 29.4%, so your estimate only just clears it. A gap that thin is smaller than the likely error in the estimate itself, which is a good reason to treat a small positive EV with scepticism.

Now run the same sum using the market's own fair probability of 27.9%. The EV is 0.2785 × £2.40 minus 0.7215 × £1, giving roughly −£0.05 per £1. That 5p loss is the margin at work: if the market's fair view is right, the draw at 3.40 loses about 5p for every pound staked over the long run. A bet only becomes attractive when your probability is meaningfully above the break-even figure, not merely above the fair one.

A sensible decision rule is to calculate EV on every price you are considering, compare it with how uncertain your inputs are, and pass on anything where the edge is smaller than that uncertainty. EV describes an average over many bets, not what will happen on one Saturday afternoon, and no model removes the chance of losing. Set a budget you can afford to lose before you place anything, and use the tools offered by UK-licensed operators, such as deposit limits, if you need them.