# How long until the next goal? The Exponential distribution

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

> Poisson counts goals. The Exponential distribution times the wait between them, and shows why "we're due a goal" isn't how probability works.

## The football question

**How long might we wait until the next goal?**

I find myself asking this all the time when watching a game, especially Celtic or Spurs.

## The concept

The [Poisson distribution](/learn/poisson-distribution) counts how many goals arrive in 90 minutes. The **Exponential distribution** looks at the same goals from the other side: the **waiting time** between them. If goals arrive at a steady average rate, the count is Poisson and the gap is Exponential.

It needs the same single number as Poisson: the rate, **λ**. A team that averages 2 goals per 90 minutes scores at

$$\lambda = \frac{2}{90} \text{ goals per minute}$$

## A football example

What is the chance we're still waiting for a goal after **30 minutes**?

$$P(T > t) = e^{-\lambda t}$$

$$P(T > 30) = e^{-\frac{2}{90} \times 30} \approx 0.513$$

Roughly a **51% chance** of no goal in the first half hour. A coin toss, for a team that averages two a game.

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

- \(T\) is how long we wait for the next goal, in minutes.
- \(t\) is the time we're asking about: 30 minutes here.
- \(\lambda t\) is how many goals we'd expect in that time: \(\frac{2}{90} \times 30 \approx 0.67\).
- \(e^{-\lambda t}\) turns that into the chance of **none** arriving: 0.513.
</div>

<figure class="rank-chart">
<div role="img" aria-label="A falling curve of the wait for a goal. The area up to 30 minutes, 49%, is the chance we've scored by then; the area beyond, 51%, is the chance we're still waiting.">

</div>
<figcaption>The wait for a goal, for a team averaging two a game. Short waits are the most common, and the curve falls away steadily. The area under the curve is probability: 49% of it lies before 30 minutes, 51% after.</figcaption>
</figure>

### The average wait isn't the typical wait

The **average** wait for a goal is \(1 / \lambda = 45\) minutes. But half of all waits are over within **31 minutes**. A few very long waits (the 0–0s) drag the average up. So "we usually score within half an hour" and "we average a goal every 45 minutes" are both true of the same team.

### The two models agree

The chance of no goal in the full 90 minutes is \(e^{-2} \approx 0.135\): exactly the 13.5% chance of a blank from [the Poisson distribution](/learn/poisson-distribution). Count the goals or time the gaps, and you get the same answer.

## "We're due a goal"

It's 0–0 after an hour. Surely something's coming?

Under this model, no more than usual. The chance of a goal in the last 30 minutes is **48.7%**, exactly the same as in the first 30. The Exponential distribution is **memoryless**: an hour of nothing doesn't make the next goal any more overdue. The same fallacy catches gamblers who think red is "due" after five blacks.

<details markdown="1">
<summary><span>Show the maths<small>Memorylessness, the mean and the median. Optional.</small></span></summary>

If we've already waited *s* minutes, the chance of waiting a further *t* is

$$\begin{aligned} &P(T > s + t \mid T > s) \\ &= \frac{e^{-\lambda (s+t)}}{e^{-\lambda s}} \\ &= e^{-\lambda t} = P(T > t) \end{aligned}$$

The past cancels out. That's memorylessness.

The mean wait is \(E[T] = 1/\lambda = 45\) minutes. The median solves \(e^{-\lambda m} = 0.5\):

$$\begin{aligned} m = \frac{\ln 2}{\lambda} &= \frac{0.693}{2/90} \\ &\approx 31.2 \text{ minutes} \end{aligned}$$

</details>

## Why it matters

Waiting times are everywhere in football:

- How long until the next goal?
- How long until the next shot?
- How long until the next card?
- How long until the next corner?

The Exponential distribution is the simplest model for all of them, and the starting point for in-play models that update win probabilities minute by minute.

## Limitations

- **The rate isn't constant through a match.** In the Scottish Premiership and Championship since 2000/01, **55.7% of goals came in the second half** and 44.3% in the first. Tiring legs, substitutions and teams chasing the game all raise the rate late on. The real "we're due a goal" feeling isn't entirely wrong; it's just not about being due. [Is a goal equally likely in any minute?](/learn/uniform-distribution) tests that minute by minute.
- **Game state changes everything.** A team that goes a goal up often sits deeper, and the team behind pushes on. One steady λ can't capture that.
- **It's a model for the next goal, not the next great goal.** Every goal counts the same, whether it's a tap-in or a 30-yard screamer.

## Try it yourself

Next time your team kicks off, note the time of each goal. After a few matches, compare how long you actually waited with the 31-minute median. Then check whether your late goals outnumber your early ones.

## Further reading

- [Seeing Theory: probability distributions](https://seeing-theory.brown.edu/probability-distributions/index.html), Brown University. Interactive: drag λ and watch the Exponential distribution change.
- [Exponential distribution: properties, proofs, exercises](https://www.statlect.com/probability-distributions/exponential-distribution), StatLect. The derivations in full, including memorylessness.
- [Exponential distribution](https://en.wikipedia.org/wiki/Exponential_distribution), Wikipedia. The formal definition and properties.