Win, draw or lose? The Multinomial distribution
A match has three possible results, not two. The Multinomial distribution handles any number of outcomes, and shows why a team's "expected" record over ten games almost never happens exactly.
Beginner Part 7 of Statistics Through Football
Contents
The football question
My team is Celtic. Over the next ten matches, what's the chance of exactly 5 wins, 3 draws and 2 defeats?
Everything in this series so far has had two outcomes: goal or miss, scored or saved. A football match has three. We need a distribution that can count more than two kinds of result.
The concept
The Multinomial distribution counts how a fixed number of attempts split across several possible outcomes, each with its own chance. It's the Binomial with more boxes:
- Binomial deals with two categories.
- Multinomial deals with more than two.
For match results it needs the number of matches, n, and a chance for each result. Say our side wins 50% of the time, draws 30% and loses 20%.
A football example
$$\begin{aligned} &P(W = 5, D = 3, L = 2) \\ &= \frac{10!}{5!\,3!\,2!} \times 0.5^{5} \times 0.3^{3} \times 0.2^{2} \end{aligned}$$
$$\begin{aligned} &= 2{,}520 \times 0.03125 \\ &\quad \times 0.027 \times 0.04 \approx 0.085 \end{aligned}$$
About an 8.5% chance.
In plain football
- \(0.5^{5} \times 0.3^{3} \times 0.2^{2}\) is the chance of one particular run, say five wins, then three draws, then two defeats.
- \(\frac{10!}{5!\,3!\,2!} = 2{,}520\) is how many different orders those same ten results could come in.
- Multiply them: any run of 5 wins, 3 draws and 2 defeats, in any order, has about an 8.5% chance.
The expected record is rarely the actual one
5-3-2 is exactly what these chances "expect" from ten games, and it's also the single most likely record. Yet it only happens about one time in twelve. The next most likely records, 6-2-2 and 6-3-1, are 7.1% each, and 4-4-2 is 6.4%. Ten matches spread across so many possible records that no single one is likely.
| Over the ten matches | Chance |
|---|---|
| Exactly 5 wins, 3 draws, 2 defeats | 8.5% |
| Unbeaten | 10.7% |
| 20 points or more | 35.9% |
| 12 points or fewer | 8.5% |
On average the side takes 18 points from these ten games. But a run of 20 or more happens more than one time in three, and a return of 12 or fewer about one time in twelve, with nothing about the team changing. That's the width of "form" you get for free.
The same idea, all over the pitch
- Match results: win, draw, loss.
- Pass types: short, long, cross.
- Shot outcomes: on target, off target, blocked.
Whenever each event falls into one of several boxes, and you want to know how a batch of them will split, it's Multinomial.
Show the mathsThe general formula, and how it collapses to the Binomial. Optional.
With n trials and m outcomes of probabilities \(p_1, \ldots, p_m\) (which add up to 1):
$$\begin{aligned} &P(X_1 = x_1, \ldots, X_m = x_m) \\ &= \frac{n!}{x_1!\,x_2!\cdots x_m!}\, p_1^{x_1} p_2^{x_2} \cdots p_m^{x_m} \end{aligned}$$
Each count on its own is Binomial: the number of wins is Binomial(n, \(p_1\)), with mean \(n p_1\). Merge draws and defeats into a single "didn't win" box and the Multinomial becomes the Binomial exactly. The chance of 5 wins from 10 is 24.6% either way.
Why it matters
Much of football data is categorical, not yes-or-no: results, pass types, shot outcomes, which third of the pitch the ball was won in. The Multinomial is the basic model for all of them. It's also a reminder that over a short run, the most likely outcome is still unlikely, so a ten-game spell far from "par" is often nothing more than the spread of possible records.
Limitations
- The chances aren't the same every match. Home or away, and the strength of the opponent, change them every week. Here they're fixed for simplicity.
- Matches are treated as independent. A run of injuries or a new manager links them in reality.
- It counts results, not scores. For scorelines, the Poisson distribution is the tool.
Try it yourself
Take your team's win, draw and loss percentages from last season. Use the formula to find the chance of their exact record over the first ten games of this season. Then check how often "par" actually happened.
Further reading
- Multinomial distribution: properties, proofs, exercises, StatLect. The derivations in full.
- Multinomial distribution, Wikipedia. The formal definition and properties.