# How many goals will we score? The Poisson distribution

Source: https://www.footballdatascience.co.uk/learn/poisson-distribution
Published: 2026-09-27

> A team averages two goals a game. How likely is it to score exactly three? The Poisson distribution turns an average into a probability for every score, and it sits underneath most football prediction models.

## The football question

**How many goals will my team score in 90 minutes?** My wife asks this a lot when we're watching Celtic or Scotland.

We don't know how many shots they'll take, so we can't count attempts the way the [Binomial distribution](/learn/binomial-distribution) counts five penalties. We only want to know how many goals might happen in a fixed stretch of time. That is the question the **Poisson distribution** answers.

## The concept

Poisson needs just one number: **λ (lambda)**, the average rate at which something happens. For a team that averages two goals a match, λ = 2 goals per 90 minutes.

From that single average, Poisson gives a probability for every possible count: 0 goals, 1 goal, 2, 3, and so on.

So instead of asking *"how many successes from a fixed number of attempts?"*, we ask *"how many times might this happen during a fixed period?"*

## A football example

Take a team with λ = 2. The chance of scoring exactly three is:

$$P(X = 3) = \frac{e^{-2} \times 2^3}{3!} \approx 0.180$$

Roughly an **18% chance of exactly three goals**.

<div class="plain" markdown="1">
In plain football

- \(\lambda = 2\): we usually score two a game.
- \(e^{-2} \approx 0.135\): the chance we draw a blank and score none at all. It is the 13.5% in the table below.
- \(2^3 = 8\): our scoring rate, counted once for each of the three goals.
- \(3! = 3 \times 2 \times 1 = 6\): the same three goals can be counted in six different orders, so we divide by 6 to count them only once.
- Put together: 0.135 × 8 ÷ 6 ≈ 0.18. We would expect to score exactly three in about one match in five and a half.
</div>

Here is the full picture for that team:

<div class="bars" markdown="1">

| Goals | Probability |
|---|---|
| 0 | 13.5% |
| 1 | 27.1% |
| 2 | 27.1% |
| 3 | 18.0% |
| 4 | 9.0% |
| 5 | 3.6% |
| 6 or more | 1.7% |

</div>

Two things stand out. A side that "averages two" fails to score more than one match in eight. And one goal is exactly as likely as two.

## From one team to a whole match

Give each side its own λ and multiply the two sets of probabilities together. That gives the chance of every scoreline.

Say the home side is expected to score 2.0 and the visitors 0.8. The model makes the home side a clear favourite: **65% to win, 21% to draw, 14% to lose**.

Yet the most likely scores are **1–0 and 2–0**, tied at only 12% each. (They tie because, for a side expected to score 2, one goal and two goals are exactly as likely as each other; the table above shows 27.1% for both.) That surprises most people. The favourite wins most of the time, but its wins are spread across 1–0, 2–0, 2–1, 3–0 and many more, so no one scoreline is very likely. It is also why exact-score predictions are so often wrong, even from good models.

## Where Poisson comes from

<details markdown="1">
<summary><span>Show the maths<small>Formulas and derivation. Optional.</small></span></summary>

Poisson is what the Binomial becomes when there are many chances, each with a small probability of success.

Imagine the 90 minutes split into *n* tiny moments, each with a probability *p* of a goal, so the average is λ = *np*. Keep λ fixed at 2 and let the moments get finer:

| Moments (n) | Binomial P(3 goals) |
|---|---|
| 10 | 0.2013 |
| 20 | 0.1901 |
| 100 | 0.1823 |
| 1,000 | 0.1806 |
| Poisson limit | 0.1804 |

In the limit the Binomial formula becomes:

$$P(X = k) = \frac{e^{-\lambda}\,\lambda^{k}}{k!}$$

A useful property: for a Poisson distribution the **mean and the variance are both λ**. That gives a quick check on real data. If goal counts vary much more than their average, a plain Poisson model is missing something.

</details>

## Why it matters

Goals are rare events scattered through a match, which is exactly the situation Poisson describes. That is why it crops up so often in football modelling.

Estimate λ for both teams, usually from their attacking strength, the opponent's defensive strength and home advantage, and you can estimate every possible scoreline. From there come win, draw and loss probabilities, over and under 2.5 goals, and both teams to score. It is where football prediction modelling starts to get interesting. [Updating a team's scoring rate](/learn/updating-scoring-rates) shows how to estimate λ as a season unfolds, and [who'll win the league?](/learn/simulating-the-league) uses it to play out whole seasons.

## Limitations

- **λ is not fixed during a match.** A team a goal up often sits deeper; a red card changes everything. Poisson assumes one steady rate for 90 minutes.
- **The two teams are treated as independent.** In reality they are not, and basic Poisson models tend to underestimate low-scoring draws such as 0–0 and 1–1. The [Dixon-Coles adjustment](/models/dixon-coles) exists to correct this, though how much it helps depends on the league.
- **Everything depends on λ.** The distribution is only as good as the average you feed it. Estimating λ well is the hard part, and this article has simply assumed it.

## Try it yourself

Here is the whole match model, set to the example above: the home side expected to score 2.0, the visitors 0.8. Find the tied 1–0 and 2–0 cells, then nudge the home side up to 2.1 and watch the tie break.

[[poisson-model]]

For over and under 2.5 goals, both teams to score and a bigger grid, open the full [Poisson Match Predictor](/models/poisson).

## Further reading

- M. J. Maher (1982), [*Modelling association football scores*](https://doi.org/10.1111/j.1467-9574.1982.tb00782.x), Statistica Neerlandica. The paper that made Poisson the standard model for football scores.
- M. J. Dixon and S. G. Coles (1997), [*Modelling association football scores and inefficiencies in the football betting market*](https://doi.org/10.1111/1467-9876.00065), Journal of the Royal Statistical Society, Series C. The low-score adjustment mentioned above.
- [Seeing Theory: probability distributions](https://seeing-theory.brown.edu/probability-distributions/index.html), Brown University. Interactive: drag λ and watch the Poisson distribution change.
- [Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution), Wikipedia. The formal definition and properties.