How the poisson distribution calculator works
Goals, corners, shots and cards arrive roughly at random around an average rate, which is what the Poisson distribution describes. Given an expected average such as xG, this calculator gives the probability of an exact count, and from it the over and under probabilities for a line.
- Take the expected average, lambda, for the period you are pricing.
- The probability of exactly k events is lambda to the power k, times e to the minus lambda, divided by k factorial.
- Add the exact probabilities from zero upwards for the probability of under a line; the rest is over.
Worked example
A team expected to score 1.5 goals.
- Exactly 2 goals: 25.1%. Exactly 0: 22.3%. Exactly 1: 33.5%.
- Under 1.5 goals: 55.8%; over 1.5: 44.2%, a fair price of 2.26 for the over.
Real scoring is slightly more spread out than Poisson, especially for draws and low totals, so treat the output as a baseline rather than a final price.
Questions
- Where does the expected average come from?
- From an expected-goals model, a team's adjusted scoring rate against this opponent, or a player's per-90 rate scaled to expected minutes. The output inherits whatever error the input has.
- Does Poisson work for player props?
- It is a reasonable baseline for counts like shots or fouls over a whole match, provided the expected minutes are right. It understates variance for players whose role changes within games.
- Why does the model underrate draws?
- Because goals are not fully independent: teams change behaviour with the scoreline. Adjusted versions correct low-scoring draws upwards; plain Poisson does not.
