The formula
P(k) = λᵏ × e^(−λ) / k! // λ = expected count in the window
P(at least 1) = 1 − P(0) = 1 − e^(−λ)
- P(0) = e^(−0.35) = 70.47%.
- P(≥1) = 1 − 70.47% = 29.53%.
- Fair price = 1 / 0.2953 → +239.
Where Poisson is a reasonable first model
- Rare, countable events in a fixed window: home runs, goals, shots in a period.
- Events that are roughly independent of each other.
- A rate that is roughly constant through the window.
Two Poissons make a soccer match
Soccer is the textbook case. Give each team an expected-goals rate for the match, treat the two goal counts as independent Poissons, and every scoreline gets a probability; summing the scoreline grid gives home, draw and away.
- P(home) = 48.9%, P(draw) = 25.4%, P(away) = 25.7%.
- The draw takes about a quarter of the probability even with a half-goal gap between the teams.
Independent Poissons are known to misprice low scores and draws, and goals change both teams' rates through score effects. Dixon–Coles corrects the low-score cells and weights recent matches; it is still a Poisson model at heart. See the Soccer Academy.
Where it breaks
| Assumption | How it fails | Symptom |
|---|---|---|
| Constant rate | A starter is pulled after 5 innings; a skater’s ice time depends on the score | Too little probability on very low and very high counts |
| Independence | Hot hands, power plays, game script that feeds the same player | Real variance larger than the mean (overdispersion) |
| Known λ | λ itself is an estimate with error | Probabilities look sharper than they deserve |
| Counts, not yards | Yards and minutes are continuous and skewed | Poisson simply does not apply |
When variance exceeds the mean, a negative binomial (a Poisson whose rate itself varies) is the usual next step. When λ is uncertain, the honest answer is a mixture of Poissons — which again spreads probability out.
A small λ error moves a home run probability a lot at long prices. At λ = 0.35 the fair price is +239; at λ = 0.30 it is +286. The Poisson math is exact; the λ never is.