Poisson distribution in football betting

The Poisson distribution is the simplest honest way to turn "this team should score about 1.5 goals" into "this team scores exactly one goal 33.5% of the time". It takes one number, a team's expected goals, and spreads it across 0, 1, 2, 3 and more. Do it for both teams, combine the two, and you have a chance for every scoreline. Most football goals markets, from correct score to clean sheets, can be priced from that grid.

The formula

For a team expected to score λ goals, the chance of exactly k goals is:

P(k) = e^−λ × λ^k / k!

Here e is about 2.718 and k! is k factorial (3! = 3 × 2 × 1 = 6). Take a home side with λ = 1.5 and an away side with λ = 1.1. These are example inputs, not a BAI8 number:

Goals	Home, λ 1.5	Away, λ 1.1
0	22.3%	33.3%
1	33.5%	36.6%
2	25.1%	20.1%
3	12.6%	7.4%
4	4.7%	2.0%
5 or more	1.9%	0.5%

The Smarkets and Pinnacle guides under sources walk through the same steps, including how to estimate λ from attack and defence strength.

From two columns to a scoreline grid

If the two teams' goals are independent, the chance of a score is the two column values multiplied. A 1–1 is 33.5% × 36.6% = 12.3%. A 1–0 is 33.5% × 33.3% = 11.1%. The six likeliest scores in this example:

Score	Chance	Fair price
1–1	12.3%	8.13
1–0	11.1%	9.01
2–1	9.2%	10.9
2–0	8.4%	11.9
0–1	8.2%	12.2
0–0	7.4%	13.5

Sum the grid in different ways and you get whole markets. Home wins add to 46.4%, draws to 25.8%, away wins to 27.8%. Three or more goals add to 48.2%. A home clean sheet is the away side's zero, 33.3%.

The markets it prices

Market	What you sum	Example chance	Explainer
Correct score	one cell	1–1: 12.3%	Correct score
Over 2.5 goals	cells with 3+ goals	48.2%	Over/under 2.5
Clean sheet	the other side's 0 row	home: 33.3%	Clean sheet
Winning margin	cells by goal difference	home by 1: 23.2%	Winning margin
Highest scoring half	the same, split by half	needs a time split	Highest scoring half

Where it goes wrong

Two assumptions do the damage. First, that the goals of the two teams are independent. Dixon and Coles found the plain model misprices the low scores, 0–0, 1–0, 0–1 and 1–1, and added a correction for them. Second, that a team's scoring rate is fixed for 90 minutes. It is not: a team that goes behind attacks, a red card changes everything, and game state moves the rate all match. And the whole grid is only as good as λ. A wrong expected-goals input gives a confidently wrong grid. Our xG explainer covers where those inputs come from.

How BAI8 relates to it

BAI8's published numbers do not come from a Poisson model. A card starts from the full match-result market with the margin removed and moves it by what the research found, as set out in how BAI8 prices a fixture. Today's card for Trabzonspor v SC Freiburg says Trabzonspor 50.7%, draw 23.2%, Freiburg 26.1%, and that is all it says. It does not publish λ for either side, so you cannot rebuild a scoreline grid from it.

What this does not tell you

Poisson does not tell you where value is. It gives you a disciplined first estimate, which you then compare with a price after removing the margin. Bookmakers can run far richer models than the plain version, so a gap between your Poisson number and the market is more often a gap in your inputs than a mistake in the price.

18+. Bet only what you can afford to lose.