How many shots until he scores? The Geometric distribution

Contents

• The football question
• The concept
• A football example
• The first shot is always the most likely, and still unlikely
• The same idea, all over the pitch
• Droughts
• Why it matters
• Limitations
• Try it yourself
• Further reading

The football question

How many shots until the striker scores his first goal?

The 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 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.

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:

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.

Show the mathsMean, cumulative probability and memorylessness. Optional.

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:

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

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 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, Brown University. Interactive: drag p and watch the Geometric distribution change.
• Geometric distribution: properties, proofs, exercises, StatLect. The derivations in full, including memorylessness.
• Geometric distribution, Wikipedia. The formal definition and properties.