How many shots until his hat-trick? The Negative Binomial distribution
The Geometric distribution waits for the first goal. The Negative Binomial waits for the third, or the fifth. It shows how long a hat-trick really takes, and it has a second job modelling goals that vary more than Poisson allows.
Intermediate Part 6 of Statistics Through Football
Contents
The football question
How many shots does a striker need to score three goals?
The Geometric distribution answered the question for the first goal. But strikers aren't judged on one goal. How long until the third? Or the fifth?
The concept
The Negative Binomial distribution gives the chance that the r-th success arrives on the k-th attempt. It needs two numbers:
- p, the chance each attempt succeeds.
- r, how many successes we're waiting for.
The Geometric is simply the special case of waiting for one success. I like this distribution because it feels like a natural extension of the last one:
- Geometric: waiting for the first success.
- Negative Binomial: waiting for repeated success.
A football example
Our striker scores with 20% of his shots. What's the chance his 3rd goal comes on exactly his 10th shot?
$$P(X = k) = \binom{k-1}{r-1}\, p^{r}\, (1 - p)^{k - r}$$
$$\begin{aligned} P(X = 10) &= \binom{9}{2} \times 0.2^{3} \times 0.8^{7} \\ &= 36 \times 0.008 \times 0.21 \\ &\approx 0.060 \end{aligned}$$
About a 6% chance.
In plain football
- The 10th shot must be the third goal, so the first 9 shots contain exactly 2 goals.
- \(\binom{9}{2} = 36\) is the number of ways to place those 2 goals among the first 9 shots.
- \(0.2^{3}\) is the three goals; \(0.8^{7}\) is the seven misses.
- Multiply: \(36 \times 0.008 \times 0.21 \approx 0.06\).
Here's the whole wait for his hat-trick:
The single most likely moment is shot 10, jointly with shot 11: both have exactly the same chance. On average he needs 15 shots, and half the time it takes 14 or more. One time in five, he's still waiting after 20 shots.
Twenty percent is a sharp finisher. At the Scottish Premiership and Championship average of about 13% of shots scored, the average wait for three goals stretches to around 23 shots.
Connected to the Binomial
The chance his third goal comes within ten shots is 32.2%. That's exactly the Binomial chance of scoring three or more from ten shots: two ways of asking the same question, one counting goals in a fixed number of shots, the other counting shots until a fixed number of goals.
The same idea, all over the pitch
- How many shots to score 3 goals?
- How many crosses to get 4 successful deliveries?
- How many tackles to win 5?
Anything that repeats with a steady chance of success, until it has succeeded a set number of times, is Negative Binomial.
Show the mathsMean, spread, and where the formula comes from. Optional.
For the r-th success to land on attempt k, attempt k must succeed and the first k − 1 attempts must contain exactly r − 1 successes. That second part is a Binomial probability, so
$$\begin{aligned} &P(X = k) \\ &= \binom{k-1}{r-1} p^{r-1}(1-p)^{k-r} \times p \\ &= \binom{k-1}{r-1}\, p^{r}\, (1 - p)^{k - r} \end{aligned}$$
Waiting for r successes is the same as r Geometric waits in a row, so the mean and variance are r times the Geometric's:
$$E[X] = \frac{r}{p} = \frac{3}{0.2} = 15$$
$$\text{sd} = \frac{\sqrt{r(1-p)}}{p} \approx 7.7$$
Its second job: goals that vary too much
Analysts also use the Negative Binomial for something else: counting goals when the numbers vary more than a Poisson distribution allows. Poisson forces the variance to equal the mean. The Negative Binomial has an extra number that lets the variance be bigger, which suits data where some matches are much wilder than others.
Does football need it? In Scottish Premiership data, raw goal counts do vary more than their average, a variance-to-mean ratio of about 1.14. But when I tested this for the SPFL Prediction Board, the extra spread turned out to come entirely from teams differing in strength, which a Poisson model with each team's own rate already captures. Match by match, goals were if anything slightly less variable than the model expected. So Poisson stays, and the Negative Binomial waits on the bench for a league that needs it.
Why it matters
Strikers, like managers, are judged over runs of games. "Only three goals in 25 shots" sounds poor, but for a 13% finisher that's close to average. The Negative Binomial tells you how long a target like a hat-trick or a ten-goal season should take, and how much luck sits either side of it.
Limitations
- The chance isn't the same every shot. A tap-in and a 25-yard effort aren't the same attempt; the model uses a single average.
- Shots aren't always independent. Confidence, fatigue and the opposition shift from one to the next.
- It counts attempts, not time. A striker who shoots twice a game reaches 15 shots in about eight matches; one who shoots four times, in four.
Try it yourself
Take a striker's shots and goals for the season. Use goals divided by shots as p, and work out how many shots his next three goals should take on average: 3 divided by p. Then watch how far the real wait lands from it.
Further reading
- Seeing Theory: probability distributions, Brown University. Interactive: set p and r and watch the Negative Binomial change.
- Negative binomial distribution, Wikipedia. The formal definition, and its use for over-dispersed counts.