# Goal or miss? The Bernoulli distribution

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

> A penalty has two outcomes and nothing in between. That simple idea, the Bernoulli trial, is the building block of expected goals and of most football statistics.

## The football question

A player steps up to take a penalty. Two outcomes. **Goal or miss. Nothing in between.**

How do we describe that with numbers?

## The concept

One event with two possible results is a **Bernoulli trial**. It needs a single number: **p**, the probability of success.

If our penalty taker scores 80% of the time:

$$P(\text{goal}) = p = 0.80$$

$$P(\text{miss}) = 1 - p = 0.20$$

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

- \(p\) is the chance the kick goes in: 0.80, or four in five.
- \(1 - p\) is everything else: saved, wide, over the bar. One in five.
- The two always add up to 1, because something has to happen.
</div>

## A football example: every shot is a Bernoulli trial

The same idea covers every shot in football: goal or no goal, each with its own **p**.

A tap-in from two yards might have p = 0.90. A speculative effort from 25 yards might have p = 0.03.

And you've already seen this idea on screen: **xG**. Expected goals estimates that probability, p, for each shot, using things like distance, angle, body part and the circumstances of the chance. Add those probabilities across all the shots a team takes and you get its match xG.

So 2.4 xG doesn't mean a team should have scored exactly 2.4 goals. It means the chances they created had a combined expected value of **2.4 goals**.

### Same xG, different match

Two teams both finish on **1.31 xG**:

| | Team A | Team B |
|---|---|---|
| Chances | A penalty (0.76) plus four scraps (0.35, 0.12, 0.05, 0.03) | Thirteen half-chances of about 0.10 each |
| Total xG | 1.31 | 1.31 |
| Chance of not scoring at all | **12.6%** | **25.1%** |

Team B is twice as likely to draw a blank. A handful of good chances is more reliable than a pile of poor ones, even when the xG totals match.

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

- A team fails to score only if **every** shot misses, so multiply the miss chances together.
- Team A: \(0.24 \times 0.65 \times 0.88 \times 0.95 \times 0.97 \approx 0.126\).
- Team B: thirteen misses in a row at 0.90 each: \(0.90^{13} \approx 0.25\).
</div>

## Visual explanation: when is a shot hardest to call?

For a Bernoulli trial, uncertainty is greatest at **p = 0.50**. A 50/50 chance is harder to predict than either an 80% or a 20% chance.

| Chance of scoring (p) | Uncertainty, p(1 − p) |
|---|---|
| 0.03 (speculative effort) | 0.03 |
| 0.20 | 0.16 |
| **0.50** | **0.25** (the maximum) |
| 0.80 (penalty) | 0.16 |
| 0.90 (tap-in) | 0.09 |

<figure class="rank-chart">
<div role="img" aria-label="An arch-shaped curve of uncertainty against the chance of scoring. It is zero at 0 and 1 and peaks at 0.5. A speculative effort at 0.03 and a tap-in at 0.90 sit low; a penalty at 0.80 and a 0.20 chance sit at the same height.">

</div>
<figcaption>How hard a shot is to call. Near-certain misses and near-certain goals sit low on the arch; a 50/50 chance sits at the top. A penalty (0.80) and a 0.20 chance are exactly as hard to call as each other.</figcaption>
</figure>

<details markdown="1">
<summary><span>Show the maths<small>Mean and variance of a Bernoulli trial. Optional.</small></span></summary>

Score a goal as 1 and a miss as 0. Then:

$$E[X] = 1 \times p + 0 \times (1 - p) = p$$

$$\begin{aligned} \text{Var}(X) &= E[X^2] - E[X]^2 \\ &= p - p^2 = p(1 - p) \end{aligned}$$

The average of a single shot is just its xG. The variance \(p(1-p)\) is a curve that is zero at \(p = 0\) and \(p = 1\), when the outcome is certain, and peaks at \(p = 0.5\).

Add independent shots together and both the means and the variances add. That is why a team's xG is simply the sum of its shots' xG values.

</details>

## Why it matters

The Bernoulli trial is the smallest building block in football statistics. Every penalty, shot, tackle, header and pass that either comes off or doesn't is one. Stack them up and you get the Binomial distribution; stretch them across 90 minutes and you get the [Poisson distribution](/learn/poisson-distribution). Expected goals is Bernoulli thinking applied to every shot in a match.

## Limitations

- **p is estimated, not known.** An xG model's p for a shot is an average over thousands of similar shots. It knows nothing about this striker's form, the keeper, or the wet ball.
- **Shots aren't always independent.** A save followed by a rebound is two shots from one moment; treating them as separate trials overstates the chance of scoring.
- **Some outcomes aren't two-way.** A shot can hit the post, win a corner or be blocked. Bernoulli deliberately ignores everything except goal or no goal.

## Try it yourself

Open [Did we deserve to win?](/models/did-we-deserve-to-win) and pick "One big chance v lots of small ones". It is the same two teams as above, as a full match: see who the chances favour, and how many points each side deserved. Then paste in the shots from a match you watched.

## Further reading

- [What is expected goals (xG)?](https://theanalyst.com/2023/08/what-is-expected-goals-xg), Opta Analyst. How the xG probability for each shot is estimated.
- [What are expected goals (xG)?](https://statsbomb.com/soccer-metrics/expected-goals-xg-explained/), StatsBomb. A second view of xG, from the company behind many public xG models.
- [Seeing Theory: probability distributions](https://seeing-theory.brown.edu/probability-distributions/index.html), Brown University. Interactive: drag p and watch the Bernoulli distribution change.
- [Bernoulli distribution](https://en.wikipedia.org/wiki/Bernoulli_distribution), Wikipedia. The formal definition and properties.