Will the third goal come before full time? The Gamma distribution

Contents

• The football question
• The concept
• A football example
• When does the third goal come?
• The first goal, the second, the third
• The same idea, all over the pitch
• Why it matters
• Limitations
• Try it yourself
• Further reading

The football question

A team averages 2 goals per 90 minutes. What's the chance their third goal arrives before full time?

The Exponential distribution told us how long until the next goal. But a comfortable win usually needs more than one. How long until the third?

The concept

The Gamma distribution gives the waiting time until a set number of events have happened, when they arrive at a steady average rate. It needs two numbers:

• k, how many events we're waiting for.
• λ, the rate they arrive at.

It's the natural extension of the Exponential:

• Exponential: waiting time until one event happens.
• Gamma: waiting time until several events happen.

A football example

Two goals per 90 minutes is a rate of

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

We're waiting for the third goal, so the time it arrives, T, is Gamma with k = 3. The chance it comes within 90 minutes is

$$\begin{aligned} P(T \le 90) &= 1 - e^{-2}\left(1 + 2 + \frac{2^2}{2}\right) \\ &= 1 - 5e^{-2} \approx 0.323 \end{aligned}$$

About a 32% chance.

So the Gamma and the Poisson distribution are two views of the same match: one counts the goals in 90 minutes, the other times how long it takes to reach three.

When does the third goal come?

Two in three matches end before the third goal comes. On average it would take 135 minutes, a match and a half, and half the time it takes more than two hours of football. Within a match, the most likely quarter of an hour for it is the last one, 75 to 90 minutes: the goals take time to build up.

The first goal, the second, the third

The same team, with the same rate, waiting for more and more goals:

Goal number	Arrives before full time
1st	86.5%
2nd	59.4%
3rd	32.3%
4th	14.3%
5th	5.3%

Each extra goal needs a longer wait, so each is less likely to fit inside the 90 minutes. A side that averages two goals a game scores five or more in about one match in nineteen.

The same idea, all over the pitch

• How long until the 3rd goal?
• How long until the 5th corner?
• How long until the 4th card?

Anything that happens at a steady rate, where you're waiting for several of them, is Gamma.

Show the mathsThe formula, its mean and spread, and the link to Poisson. Optional.

For k events at rate \(\lambda\), the waiting time T has density

$$f(t) = \frac{\lambda^k t^{k-1} e^{-\lambda t}}{(k-1)!}$$

When k is a whole number, the chance of waiting no more than t comes straight from the Poisson: the k-th event has arrived by time t exactly when at least k events have happened by then.

$$P(T \le t) = 1 - \sum_{j=0}^{k-1} \frac{e^{-\lambda t} (\lambda t)^j}{j!}$$

Waiting for k events is k Exponential waits in a row, so

$$E[T] = \frac{k}{\lambda} = \frac{3}{2/90} = 135 \text{ min}$$

$$\text{sd} = \frac{\sqrt{k}}{\lambda} \approx 78 \text{ min}$$

The single most likely moment is \((k-1)/\lambda = 90\) minutes. The Gamma is to the Exponential what the Negative Binomial is to the Geometric: the same idea, measured in time instead of attempts.

Why it matters

"Will they score three?" is one of the most asked questions in football, from the pundit's prediction to the over 2.5 goals market. The Gamma frames it as a race against the clock. It also explains why a strong side that "should" score three rarely gets there before the last quarter of an hour: they're usually still waiting.

Limitations

• Goals don't arrive at a steady rate. More come in the second half than the first, as the Exponential article shows, so the third goal leans even later than this.
• Game state changes the rate. A team two goals up often eases off; one chasing the game pushes on.
• Stoppage time is ignored. Real matches run past 90 minutes, which gives a few more minutes for the third goal to arrive.

Try it yourself

Take your team's average goals per game. Use the Poisson chance of 0, 1 or 2 goals to work out how often their third should come before full time. Then count how often they actually scored three or more last season.

Further reading

• Gamma distribution: properties, proofs, exercises, StatLect. The full derivations.
• Gamma distribution, Wikipedia. The formal definition, and its links to the Exponential and Poisson.