Goal or miss? The Bernoulli distribution

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.

27 September 2026 Beginner Part 1 of Statistics Through Football

Contents

The football question
The concept
A football example: every shot is a Bernoulli trial
Same xG, different match
Visual explanation: when is a shot hardest to call?
Why it matters
Limitations
Try it yourself
Further reading

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:

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

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

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.

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.

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: $0.90^{13} \approx 0.25$.

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

Show the mathsMean and variance of a Bernoulli trial. Optional.

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.

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. 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? 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.