Many player props count events: strikeouts, shots on goal, threes made, receptions, saves. The Poisson distribution is a simple, surprisingly good model for count stats, and it lets you turn a projected average into fair odds for any line.
What Poisson does
If an event happens independently at an average rate λ (lambda), Poisson gives the probability of exactly k events:
Add up the probabilities to get "over" and "under".
Worked example: strikeouts
A pitcher projects for λ = 6.2 strikeouts. Line: 5.5.
| Strikeouts | Probability |
|---|---|
| 0–5 combined | 41.4% |
| 6+ (over 5.5) | 58.6% |
Fair odds for the over: about −141. If a book offers −115 (53.5% implied), the over has an edge of 0.586 × 1.870 − 1 ≈ +9.6%.
At a line of 6.5, P(7+) = 42.6%, fair about +135.
Use the Poisson calculator to do this for any average and line.
Backing out λ from a sharp line
Often you do not have a projection, but you do have a sharp prop: Over 6.5 at a fair 45%. Find the λ that gives P(7+) = 45% (about 6.35), then use it to price a different line at another book, like 5.5 or 7.5. This is how bettors compare props with different numbers. (How to find +EV player props.)
Where Poisson works well
- Strikeouts (with a sensible innings projection)
- Shots on goal (NHL and soccer)
- Goals in soccer and hockey
- Threes made, steals, blocks
- Receptions (reasonably)
Where it struggles
- Yards and points, which are continuous-ish and lumpy (a single long touchdown); normal or empirical distributions work better
- Overdispersed stats, where variance is bigger than the mean (e.g., a player whose minutes vary a lot). A negative binomial fits better.
- Game-script dependence: blowouts change minutes and usage
- Correlation: a pitcher's strikeouts depend on how long he stays in, which depends on how he pitches
When variance is higher than the mean, Poisson overstates the chance of results near the average and understates the tails. Unders on high lines and overs on low lines may look different under a better model.
Practical tips
- Use Poisson to compare nearby lines for the same player, where its errors mostly cancel.
- Calibrate λ from a sharp line, not just a season average.
- Adjust λ for minutes or innings, opponent and pace.
- Treat small edges (under 3%) skeptically; model error is larger than that. (Minimum edge.)
Frequently asked questions
What is the Poisson distribution used for in betting?
To estimate the probability of a given number of events (strikeouts, goals, shots) from an average rate, which converts to fair odds for over/under lines.
Is Poisson accurate for player props?
It is a good approximation for count stats with stable opportunity, and weaker for yards, points or players with variable minutes.
How do I calculate a Poisson probability?
P(k) = e^(−λ) × λ^k ÷ k!. Sum the probabilities above the line for the over, or use a Poisson calculator.