# How many shots until he scores? The Geometric distribution

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

> A striker scores with one shot in five. How many shots until his first goal? The Geometric distribution answers it, and shows why a three-match drought is often just bad luck.

## The football question

**How many shots until the striker scores his first goal?**

The [Exponential distribution](/learn/exponential-distribution) measures the wait for a goal in minutes. Here we count attempts instead.

## The concept

The **Geometric distribution** gives the chance that the first success comes on the *k*-th attempt. Like the [Bernoulli trial](/learn/bernoulli-distribution) it's built from, it needs just one number: **p**, the chance each attempt succeeds.

- Exponential: **how long** until the next event.
- Geometric: **how many attempts** until the first success.

## A football example

Our striker scores with **20%** of his shots, so p = 0.20. What's the chance his first goal comes on exactly his **4th shot**?

$$P(X = k) = (1 - p)^{k - 1}\, p$$

$$P(X = 4) = 0.8^{3} \times 0.2 = 0.1024$$

About a **10% chance**.

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

- \(0.8^{3}\): his first three shots all miss. Each miss has a 0.8 chance, so three in a row is \(0.8 \times 0.8 \times 0.8 = 0.512\).
- \(0.2\): then the fourth goes in.
- Multiply them: \(0.512 \times 0.2 \approx 0.10\).
</div>

Twenty percent is a sharp finisher, by the way. In the Scottish Premiership and Championship, about **13%** of all shots are goals (and about 30% of shots on target).

Here's his whole first-goal picture:

<div class="bars" data-col="2" markdown="1">

| First goal on shot | Probability | Scored by then |
|---|---|---|
| 1 | 20.0% | 20.0% |
| 2 | 16.0% | 36.0% |
| 3 | 12.8% | 48.8% |
| 4 | 10.2% | 59.0% |
| 5 | 8.2% | 67.2% |
| 6 | 6.6% | 73.8% |

</div>

### The first shot is always the most likely, and still unlikely

The single most likely shot for his first goal is **shot 1**. It always is, for any p. Yet even after four shots, there's still a 41% chance he hasn't scored. On average he needs **5 shots**.

## The same idea, all over the pitch

- First goal on the 4th shot.
- First successful tackle on the 3rd attempt.
- First completed long pass on the 2nd try.

Anything that repeats with the same chance of success until it comes off is Geometric.

## Droughts

Say our 20% striker takes three shots a game. The chance he goes **three matches without scoring**, nine shots and no goal, is \(0.8^{9} \approx\) **13%**. Roughly one run of three games in eight, with his finishing exactly as good as ever.

The Geometric distribution is **memoryless**, like the Exponential. After nine misses, the chance his next shot goes in is still 20%. He isn't "due" one, and he hasn't lost it either. Before a manager drops a striker for a drought, it's worth checking whether the numbers are simply doing what numbers do.

<details markdown="1">
<summary><span>Show the maths<small>Mean, cumulative probability and memorylessness. Optional.</small></span></summary>

The first success comes by attempt *k* unless the first *k* all fail:

$$P(X \le k) = 1 - (1 - p)^{k}$$

The mean number of attempts is

$$E[X] = \frac{1}{p} = \frac{1}{0.2} = 5$$

And the past doesn't matter:

$$\begin{aligned} &P(X > s + t \mid X > s) \\ &= \frac{(1-p)^{s+t}}{(1-p)^{s}} \\ &= (1-p)^{t} = P(X > t) \end{aligned}$$

</details>

## Why it matters

Strikers are judged in streaks: "hasn't scored in five", "can't stop scoring". The Geometric distribution gives a baseline for how long a perfectly consistent player should expect to wait. Only a drought much longer than that baseline is real evidence that something has changed.

## Limitations

- **p changes from shot to shot.** A tap-in and a 30-yard effort aren't the same attempt. A striker's 20% is an average over very different chances; [expected goals](/learn/bernoulli-distribution) gives each shot its own p.
- **Shots aren't always independent.** Confidence, fatigue and the opposition all shift from one attempt to the next.
- **It only counts to the first success.** For "how many shots until his third goal?", we need the next distribution in the series.

## Try it yourself

Pick a striker and find his shots and goals for the season. Use goals divided by shots as p, work out \((1 - p)\) raised to his shots per game times three, and you have the chance of a three-match drought by luck alone. Then compare it with how often he actually goes three games without scoring.

## Further reading

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