Is that a feat, or a coincidence?

James had one probability: his pitcher's team won 52% of its games, every game. Under a constant chance, the number of successes in n independent tries is the binomial distribution, and his factorial formula is exactly its probability mass function. Our matches do not share a chance. Liverpool at home to a promoted side and Liverpool away at Arsenal are not the same trial, and the five-feature logistic draw model, recalibrated by its isotonic map gives each of them its own figure. A sum of independent coin flips with different chances is not binomial — it is the Poisson-binomial distribution, and that is what every number on this page uses.

Measured, not assumed — and the honest reading is that the two rows disagree about how much this matters. Over a whole career the correction is worth about one percent, because draw chances live in a narrow band between roughly 7% and 40% and p(1−p) is nearly flat across it. Anyone claiming this page needs the Poisson-binomial because the binomial's spread is wrong would be overselling it. Over a single run it is worth far more, and always in the same direction: the binomial makes a record look more ordinary than it was.

The real reason for it is neither. The expectation is only available per match. Comparing a team against the sum of its own fixtures' chances — what these matches deserved — is the entire point, and once the average is per-match, the matching distribution is this one. Putting a binomial beside a per-match average is mixing two models.

For each run: how many draws the model expected across those exact fixtures, the chance that none of them landed, and what a binomial at the run's own average chance would have said instead. Every figure in this table is asked of a run that was selected for being long, which overstates it. The table underneath is the version that is not.

Asking how unlikely this run was, after picking it for being the longest, is asking the wrong question. The right one is how many runs of at least that length ought to exist somewhere in the database at all. That is worked out by walking every position a run could start from — every match after a draw, and every match after a gap in coverage — and adding up the chance that a run starts there and survives.

Expectations add whether or not the matches are independent, so this column does not lean on that assumption at all — only the spreads and the tails above it do.

Watch which way the flat column goes, because it reverses. It expects more short runs than the model and fewer long ones. Two different effects are at work and they point opposite ways. Inside a single run, averaging the chances first always makes the run look more likely — that is the concavity from the table above. Across the whole database, giving every team the same chance makes a long run far less likely, because a long run is overwhelmingly produced by the teams that hardly ever draw, and a flat rate deletes them. The second effect is the bigger one at the top of this table: teams are not alike, and a league in which they were would almost never produce a fifty.

Ordered by how far each sits from what the model expected of its own fixtures. The adjusted column is what makes an ordered table defensible at all: test this many subjects and a handful will clear any threshold on noise alone. Read it as "if I call every row above this one interesting, this is roughly the share that is not". Note that the top of this table has adjusted values nowhere near small — which is the finding, not a disappointment.

Ordered by how far each sits from what the model expected of its own fixtures. The adjusted column is what makes an ordered table defensible at all: test this many subjects and a handful will clear any threshold on noise alone. Read it as "if I call every row above this one interesting, this is roughly the share that is not". Note that the top of this table has adjusted values nowhere near small — which is the finding, not a disappointment.

Matches are not quite independent.

The distribution assumes they are, given their chances. They are not: the same squad, the same manager, the same injury list, and a side that starts playing for draws keeps doing it. The model's per-match figure absorbs much of that — it is built from recent form — but the leftover correlation makes the true spread a little wider than the one quoted here, so the tails above are, if anything, slightly too small. No fudge factor has been applied to hide that.

A small p-value is not skill.

It says the model did not expect this. For us that is as likely to be a gap in the model as a property of the team, and when the model is wrong that is a finding about the model.

The probabilities are in sample.

Every played match here was part of the five-feature logistic draw model, recalibrated by its isotonic map's own training set, so its chance was fitted already knowing the result. That flatters the model, and every figure built on it inherits the flattery. football:report:draw-holdout is the standing out-of-sample check.

How the tails were computed.

Exactly, by folding one match into the distribution at a time — no factorials, which lose every digit they have long before 700! overflows. Above 1,000 matches a skew-corrected normal approximation is used instead and the row says so; measured against the exact answer it is within 0.006 in the tails. Errors in the exact path accumulate at about 1e-13 over a thousand matches.

Draws are ninety-minute draws.

As everywhere on this site: a shoot-out win stays a draw.

Players and managers are missing.

The striker version of this question — "were those goals him, or the side around him?" — needs to know who was on the pitch, and there are no per-match appearances in this database. Managers have no source at all. Neither is being guessed at.

Probabilities from the five-feature logistic draw model, recalibrated by its isotonic map. Last measured 06 Nov 2026. Descriptive statistics over historical results, not betting advice.