Poisson distribution in betting

The Poisson distribution models the number of rare, independent events in a fixed period — which is a fair description of goals in a football match.

Feed it one number, the expected goals for a team, and it returns the probability of exactly zero, one, two, three or more goals. Combine both teams and you have a full scoreline matrix, from which every goals market can be derived: match odds, totals, BTTS, correct score. Its weakness is the independence assumption — real matches have game state, red cards and late collapses that a pure Poisson model does not know about.

P(k) = (λ^k x e^-λ) / k!

With λ = 1.4, P(0 goals) = e^-1.4 = 24.7%, P(1) = 34.6%, P(2) = 24.2%. If the away side has λ = 1.0, multiplying the grids gives P(1-1) = 34.6% x 36.8% = 12.7% — a fair correct-score price of 7.87 built from two numbers.

It is the entry point to building your own prices rather than reacting to someone else's. Once you can produce a scoreline matrix, every football market becomes checkable.

Applying Poisson to NPFL fixtures requires local λ estimates, not imported European averages. Domestic scoring rates are lower, so a model calibrated on the Premier League will systematically overprice Overs.

• Assuming home and away goals are independent when they clearly interact in one-sided matches.
• Using season averages instead of opponent-adjusted expected goals.
• Trusting the tail of the distribution, where Poisson underestimates high-scoring blowouts.