Form or luck?
"He's lost his touch." "He's on fire." Take a striker whose ability never changes, not for a single match, and see how often luck alone produces the droughts and streaks we call form.
Scores in of matches.
Try an example
In a season, with nothing about the striker changing, the chance of at least one run of :
- Without scoring
- Scoring in every one
Longest run without scoring, across the season
The most likely longest drought is games. Pale bars are droughts at least as long as the run you're asking about.
How it works
A striker averaging \(g\) goals a game scores at least once in a match with probability
$$q = 1 - e^{-g}$$Each match is then a Bernoulli trial: he scores or he doesn't, with the same chance every time. The model follows the season match by match, tracking how long the current run of blanks is, and adds up exactly how likely every longest drought and longest streak is. Nothing is simulated.
In plain football
- \(g\) is his average goals per game: 0.5 is one every two matches.
- \(e^{-g}\) is the chance of a blank, from the Poisson distribution. So \(q\) is the chance he scores at least once.
- The striker is identical in every match. Any drought or streak here is luck.
What it shows
Try "Good, 0.5 a game". A striker who scores every other match on average goes five or more games without scoring at some point in three seasons out of four, and five games is his most likely longest drought. A prolific striker at 0.8 a game puts together a five-match scoring streak in most seasons, again with nothing changing. Some real form exists, but these runs are what a constant player produces anyway. The Geometric distribution article works through the same idea, shot by shot.
Where it goes wrong
- Real strikers miss matches, come off the bench, and sometimes face a run of strong opponents. A cluster of hard fixtures can produce a drought that is neither form nor pure luck.
- It can't tell you form doesn't exist. It tells you how much of what looks like form a constant player produces on his own, which is the bar real form has to clear.
These probabilities come from a statistical model and are for analysis and education. Football remains uncertain and model predictions will frequently be wrong.