Poisson distribution betting uses one number per side, its expected goals, to work out the chance of every scoreline, then adds those chances into fair prices for the result, total goals and correct score markets. With illustrative expected goals of 1.6 for the home side and 1.1 for the away side, the model gives the home win 48.96%, over 2.5 goals 50.64% (a fair price of $1.97) and 1-1, at 11.83%, as the likeliest single score.
The hard parts are the two inputs and the model's blind spots: it treats the sides' goals as independent, so it undercounts draws, and it fits some sports far better than others. It turns goal expectations into the fair prices a betting strategy compares offers against, and how to build a betting model covers the wider steps.
The Poisson formula in one line
The Poisson distribution gives the chance of a count, such as one side's goals in a soccer match, from its average. In plain text:
P(k goals) = e^-l x l^k / k!
Here l is the side's expected goals (statisticians write it as the Greek letter lambda), e is the constant 2.718, and k! means 1 x 2 x ... x k, with 0! counted as 1. For the home side at 1.6:
- P(0) = e^-1.6 = 0.2019, or 20.19%
- P(1) = 0.2019 x 1.6 = 32.30%
- P(2) = 0.2019 x 1.6 x 1.6 / 2 = 25.84%
In Excel or Google Sheets, POISSON.DIST(2, 1.6, FALSE) returns the same 25.84%. Every count for both sides, with each column adding to 100% give or take rounding:
| Goals | Home side, 1.6 expected | Away side, 1.1 expected |
|---|---|---|
| 0 | 20.19% | 33.29% |
| 1 | 32.30% | 36.62% |
| 2 | 25.84% | 20.14% |
| 3 | 13.78% | 7.38% |
| 4 | 5.51% | 2.03% |
| 5 or more | 2.37% | 0.54% |
Where the two expected goals come from
A common starting point scales the league's average by the two teams' strengths:
home expected goals = league home average x home side's attack x away side's defence
Attack and defence are ratios to the league average: an attack of 1.25 scores 25% more than an average side in that role, and a defence of 0.88 concedes 12% fewer. With illustrative figures, 1.81 home goals a game against a 1.45 home average is a home attack of 1.81 / 1.45 = 1.248, about 1.25, so:
- home side: 1.45 league home average x 1.25 home attack x 0.88 away defence = 1.595, about 1.6
- away side: 1.15 league away average x 1.05 away attack x 0.91 home defence = 1.099, about 1.1
Ratings built from expected goals (xG), the quality of the chances each side creates and allows, are steadier over a handful of games than ratings from goals alone. The market's own inputs can be read off its total goals line and handicap, as over/under goals betting shows.
The sum and the split do different jobs. Over and under lines depend only on the total, 1.6 + 1.1 = 2.7, because two independent Poisson counts add up to a Poisson count with the summed average. The result depends on the split:
| Expected goals, home and away | Total | Over 2.5 goals | Home win | Draw | Away win |
|---|---|---|---|---|---|
| 1.6 and 1.1 | 2.7 | 50.64% | 48.96% | 24.89% | 26.15% |
| 1.9 and 0.8 | 2.7 | 50.64% | 63.34% | 21.54% | 15.12% |
Worked example: 1.6 against 1.1 expected goals
The scoreline grid
Treating the two sides as independent, a score's chance is the home side's chance of its goals times the away side's chance of its goals. For 2-1: 25.84% x 36.62% = 9.46%. Scores are written home first:
| Home goals | Away 0 | Away 1 | Away 2 | Away 3 | Away 4 |
|---|---|---|---|---|---|
| 0 | 6.7% | 7.4% | 4.1% | 1.5% | 0.4% |
| 1 | 10.8% | 11.8% | 6.5% | 2.4% | 0.7% |
| 2 | 8.6% | 9.5% | 5.2% | 1.9% | 0.5% |
| 3 | 4.6% | 5.0% | 2.8% | 1.0% | 0.3% |
| 4 | 1.8% | 2.0% | 1.1% | 0.4% | 0.1% |
These 25 cells hold 97.1% of the chance; the rest sits in scores where one side gets five or more. Cells below the diagonal make the home win, the diagonal the draw and cells above it the away win, so extend the grid in a spreadsheet to lose nothing.
Fair prices for the result, total and correct score
Fair price = 1 / chance, before any bookmaker margin, worked from the unrounded chance:
| Market | Chance | Fair price |
|---|---|---|
| Home win | 48.96% | $2.04 |
| Draw | 24.89% | $4.02 |
| Away win | 26.15% | $3.82 |
| Over 2.5 goals | 50.64% | $1.97 |
| Under 2.5 goals | 49.36% | $2.03 |
| Both teams to score | 53.24% | $1.88 |
The total needs no grid: under 2.5 = e^-2.7 x (1 + 2.7 + 2.7 x 2.7 / 2) = 0.0672 x 7.345 = 49.36%, so over 2.5 = 50.64%. Both teams to score = (1 - e^-1.6) x (1 - e^-1.1) = 0.7981 x 0.6671 = 53.24%, and both teams to score covers that market. The likeliest correct scores:
| Score | Chance | Fair price |
|---|---|---|
| 1-1 | 11.83% | $8.45 |
| 1-0 | 10.75% | $9.30 |
| 2-1 | 9.46% | $10.57 |
| 2-0 | 8.60% | $11.62 |
| 0-1 | 7.39% | $13.53 |
| 0-0 | 6.72% | $14.88 |
| 1-2 | 6.51% | $15.37 |
Correct score betting sets these prices against a bookmaker's full market, where any other score collects the scores the list leaves out.
Comparing the model with bookmaker prices
Say Bookmaker A offers over 2.5 at $2.08 and under at $1.78 (illustrative). Its book adds to 1 / 2.08 + 1 / 1.78 = 48.08% + 56.18% = 104.26%, and against your 50.64%:
expected value per $1 = chance x price - 1 = 0.5064 x 2.08 - 1 = +5.3%
That edge rests on the 2.7 total being right. Move it a quarter of a goal either way:
| Expected total | Under 2.5 | Over 2.5 | Fair over price | Expected value per $1 at $2.08 |
|---|---|---|---|---|
| 2.45 | 55.67% | 44.33% | $2.26 | -7.8% |
| 2.70 | 49.36% | 50.64% | $1.97 | +5.3% |
| 2.95 | 43.45% | 56.55% | $1.77 | +17.6% |
A quarter-goal error turns a +5.3% bet into a -7.8% one. With the margin taken out, Bookmaker A's market rates over 2.5 at 48.08 / 104.26 = 46.1%, nearer the 2.45 row than yours. When the two disagree by that much, look for what the market knows, such as a weakened line-up, before treating the gap as value.
Risk: Betting involves risk. A Poisson price is an estimate built on two estimates, a +EV bet can still lose, and often does, and there is no guarantee of profit. See responsible gambling for limits and support.
Where Poisson goes wrong: draws and correlation
The plain model assumes both sides score independently, at a steady rate, all match. Real matches break both assumptions:
- Low scores and draws. Mark Dixon and Stuart Coles, modelling English league and cup football in a 1997 paper, found 0-0 and 1-1 turned up more often than independent Poisson counts allow.
- Game state. Once a goal goes in, the leading side may sit deeper while the other pushes forward, and a red card changes the rate again.
- Spread. A Poisson count's variance equals its average, so if a side's goals per game over a season vary well beyond that, the model understates heavy wins and blanks.
The Dixon and Coles adjustment multiplies the four lowest scores by factors built from a small dependence figure, r, fitted to past results; it is negative when low-scoring draws are undercounted. With home expected goals h and away expected goals a: 0-0 is multiplied by 1 - h x a x r, 1-0 by 1 + a x r, 0-1 by 1 + h x r, and 1-1 by 1 - r.
Example: With an illustrative r of -0.10 at 1.6 and 1.1, 0-0 is multiplied by 1.176, 1-0 by 0.89, 0-1 by 0.84 and 1-1 by 1.10, and each of the four moves by about 1.18 points.
| Market | Plain Poisson | Adjusted |
|---|---|---|
| 0-0 | 6.72% | 7.90% |
| 1-0 | 10.75% | 9.57% |
| 0-1 | 7.39% | 6.21% |
| 1-1 | 11.83% | 13.01% |
| Draw | 24.89% ($4.02) | 27.26% ($3.67) |
| Home win | 48.96% ($2.04) | 47.77% ($2.09) |
| Away win | 26.15% ($3.82) | 24.97% ($4.01) |
| Over 2.5 goals | 50.64% | 50.64% |
The draw gains 2.37 points and its fair price shortens from $4.02 to $3.67, while over 2.5 does not move, because all four adjusted scores have two goals or fewer. A plain model can price the total well and the draw badly at the same time, so fit r to the league you bet on rather than borrowing a figure. Two mistakes, costed on the numbers above:
| Mistake | What it costs |
|---|---|
| Inputs taken from goals over a few games | A quarter-goal error in the total moves the fair over 2.5 price from $1.97 to $2.26 or $1.77 |
| Pricing the result off the plain model | A home win at $2.08 shows 0.4896 x 2.08 - 1 = +1.8% on the plain model and 0.4777 x 2.08 - 1 = -0.6% once draws are adjusted |
Poisson in other sports
Poisson suits counts of fairly rare, separate events, and it suits large totals far less:
| Sport | What you count | How well Poisson fits | What to add or use instead |
|---|---|---|---|
| Ice hockey | Goals per side | A reasonable start, as in soccer | An allowance for late empty-net goals, once a trailing side pulls its goalkeeper, and a check on whether the market includes overtime |
| NRL | Tries per side or per player | Reasonable for try counts | A bell-curve (normal) model for margins and total points |
| AFL | One player's goals | Reasonable for one player | A normal model for team totals and margins, which are large and built from scores of 6 and 1 |
| NBA | One player's threes made | Fair for a small count | A normal model for points totals |
| Tennis | Points, games and sets | Poor: the scoring structure decides results | A model built on each player's chance of winning a point on serve |
For one player, the chance of at least one is 1 - e^-m, where m is the expected count. An AFL forward expected to kick 1.3 goals (illustrative) kicks at least one with a chance of 1 - e^-1.3 = 1 - 0.2725 = 72.75%, a fair $1.37. NRL try scorer betting and NBA player props price related player markets.
Checking a Poisson price on the Terminal
B337 has no model, no ratings and no selections, so the Poisson price is yours. The Terminal's sports board shows bookmakers side by side on head to head, line and total markets, and player props where bookmakers offer them. Its EV overlays take the margin out of a global market reference to give a second, market-based estimate to set beside your model's. Both are estimates.
Prices are read on a repeating cycle, so a price on screen can trail the bookmaker's own; confirm it in your bookmaker account before you bet. A free account opens a limited view of the Terminal with live odds; EV overlays start with Terminal View.
For free and confidential support call 1800 858 858 or visit gamblinghelponline.org.au.
Risk: Betting involves risk. Two estimates that agree do not make a bet certain, and a price can move before your bet is accepted. See responsible gambling for limits and support.