Poisson Distribution explained for betting

In one sentence

The Poisson distribution tells you how likely each count (0, 1, 2, 3 goals and so on) is when events happen randomly at a known average rate.

How it works

Imagine a match as 90 minutes of small chances, each with a tiny probability of becoming a goal. If you know the average number of goals a team should score, the Poisson distribution spreads that average across every possible scoreline count. You only need one number, the expected goals, often written λ (lambda).

Football suits it reasonably well because goals are rare, fairly independent of each other, and happen at a roughly steady rate. Give each team its own λ, work out the probability of each goal count, and multiply the two teams together to get a full correct score grid. From that grid you can price match odds, over/under, both teams to score and correct score.

The same idea works for any count of rare events: corners, cards, tries, wickets in a spell, or even the number of matched bets arriving in a minute.

The maths

P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}

X is the number of events (goals) in the match.
k is the specific count you are asking about (0, 1, 2 ...).
λ is the expected number of events, which is both the mean and the variance.
e is the constant 2.718..., and k! means k × (k − 1) × ... × 1.

In words: the chance of exactly k goals depends only on the average rate, and the spread is fixed once you pick the average.

Worked betting example

Over/Under goals, illustrative inputs. Your model gives the home side 1.6 expected goals and the away side 1.1.

Home goals from the Poisson formula: 0 goals 20.19%, 1 goal 32.30%, 2 goals 25.84%, 3 goals 13.78%.
If the two teams score independently, total goals are also Poisson with λ = 1.6 + 1.1 = 2.7.
Probability of 0, 1 or 2 total goals (under 2.5) = 49.36%, so over 2.5 = 50.64%.
Fair odds for over 2.5 = 1 ÷ 0.5064 ≈ 1.97. Fair odds for under 2.5 ≈ 2.03.
The exchange offers 2.10 on over 2.5. Backing £10: expected value = 0.5064 × £11 − 0.4936 × £10 ≈ +£0.63.
After 2% commission on winnings: 0.5064 × £11 × 0.98 − 0.4936 × £10 ≈ +£0.52.

That edge is thin and depends entirely on your λ values being right. Nudge either team's expected goals by 0.1 and the answer moves noticeably.

Where it's good

Quick pre-match pricing of over/under, correct score and both teams to score in football.
Counting markets such as bookings, corners (as a first pass) and tries.
In-play: scale λ by the minutes remaining to reprice totals as the clock runs down.
A baseline that more complex models must beat before you trust them.

Limitations and pitfalls

Variance equals the mean. Real corner and card counts are usually more spread out, so the tails get underpriced (see negative binomial).
Independence between the two teams is false in practice: low-scoring draws such as 0-0 and 1-1 occur more often than the basic model says.
Scoring rates change within a match. Red cards, a team chasing the game and late pressure all break the constant-rate idea.
The model is only as good as λ. Most of the work, and most of the error, is in estimating expected goals, not in the formula.
Liquid football markets on Betfair are efficient. A plain Poisson model built from season averages rarely beats the closing price.
Check value after 2% commission, not before.

How to build it

scipy.stats.poisson in Python (pmf, cdf, sf) or dpois in R handle the maths.
penaltyblog and statsmodels (Poisson GLM) can fit attack and defence ratings from past results.
Data: several seasons of results or, better, expected goals by match, with home and away split.
Tip: weight recent matches more heavily; team strength drifts through a season.

Dixon-Coles: fixes Poisson's low-score draw problem.
Bivariate Poisson: lets the two teams' goals be correlated.
Negative binomial: for counts more spread out than Poisson allows.
Skellam: the goal difference between two Poisson teams.
Expected goals: the usual source of λ.