The Poisson Model: Assumptions and Formula
At its core, the Poisson distribution models how many times an event occurs in a fixed interval (a game, a start, a period) when certain conditions are approximately true. Understanding these conditions helps you know when Poisson will work—and when it won't.
The Four Poisson Assumptions
The Poisson distribution works when four conditions are approximately true:
1. Events Are Discrete Counts
Outcomes must be whole numbers: 0, 1, 2, 3, and so on. You can't score 1.7 touchdowns or record 2.3 strikeouts.
2. Events Are Independent
One occurrence doesn't meaningfully change the chance of the next. A receiver's first touchdown shouldn't dramatically affect his probability of scoring a second (though in reality, game script may shift).
3. Events Occur at a Roughly Constant Average Rate
The expected rate (λ) stays relatively stable over the interval. A pitcher's strikeout rate shouldn't vary wildly from inning to inning under normal circumstances.
4. Two Events Don't Occur at the Exact Same Instant
This is a technical condition that's usually reasonable for sports counting stats—a player can't score two touchdowns on the exact same play.
Note
In prop-betting terms, λ (lambda) is your projected average rate (expected count), and k is the specific count you're evaluating (exactly 0, exactly 2, etc.).
The Poisson Probability Formula
The Poisson probability mass function (PMF) is:
Poisson Probability
P(X = k) = (λ^k × e^(-λ)) / k!=POISSON.DIST(k, λ, FALSE)Where:
- P(X = k) = Probability of exactly k events occurring
- λ = Expected number of events (your projection)
- k = Specific number of events you're calculating probability for
- e = Mathematical constant ≈ 2.71828
- k! = k factorial (k × (k-1) × (k-2) × ... × 1)
You typically won't compute this by hand—Excel will do it instantly. But understanding what it means matters: the distribution is bounded at 0 and often right-skewed for small λ values.
Excel Functions for Poisson
Here are the key Excel patterns you'll use constantly:
Exactly k Events
P(X = k)=POISSON.DIST(k, λ, FALSE)k or Fewer Events (Cumulative)
P(X ≤ k)=POISSON.DIST(k, λ, TRUE)More Than k Events
P(X > k) = 1 - P(X ≤ k)=1 - POISSON.DIST(k, λ, TRUE)Practical Excel Examples
| What You Want | Excel Formula |
|---|---|
| P(exactly 2 TDs) with λ=1.5 | =POISSON.DIST(2, 1.5, FALSE) |
| P(0 or 1 TDs) with λ=1.5 | =POISSON.DIST(1, 1.5, TRUE) |
| P(2 or more TDs) with λ=1.5 | =1-POISSON.DIST(1, 1.5, TRUE) |
| P(Under 0.5 TDs) = P(0 TDs) | =POISSON.DIST(0, λ, TRUE) |
| P(Over 0.5 TDs) = P(1+ TDs) | =1-POISSON.DIST(0, λ, TRUE) |
| P(Over 7.5 Ks) = P(8+ Ks) | =1-POISSON.DIST(7, λ, TRUE) |
| P(Under 7.5 Ks) = P(≤7 Ks) | =POISSON.DIST(7, λ, TRUE) |
Tip
For half-point lines (0.5, 1.5, 7.5, etc.), there are no pushes, so Over and Under are perfect complements. Use INT(Line) to convert a half-point line into the correct integer cutoff.
How the Distribution Changes with λ
The shape of the Poisson distribution depends entirely on λ:
Small λ (0.5 - 1.5): Common for TDs, HRs
- Right-skewed
- 0 is often the single most likely outcome
- Long right tail for occasional spike games
Medium λ (2 - 5): Common for Strikeouts, 3s
- Less skewed
- Peak moves rightward
- More probability mass in middle outcomes
Large λ (10+): Approaches Normal
- Nearly symmetric
- Can use normal approximation
- Rule of thumb: When λ > 20, use Normal with μ = λ and σ = √λ
Key Insight
As λ increases, the distribution shifts right and becomes less skewed. For rare events (small λ), 0 is often the single most likely outcome.
Try the Poisson Calculator
Use the calculator below to experiment with different λ values and see how the probabilities change:
Poisson Calculator
Try the interactive calculator for this concept
📝 Exercise
Instructions
Practice using the Poisson distribution formula and Excel functions.
A running back has λ = 0.8 touchdowns per game. Using Poisson, what's the approximate probability he scores exactly 0 touchdowns?
You want to calculate P(Over 2.5 strikeouts) with λ = 5.0. Which Excel formula is correct?
Which of the following violates a key Poisson assumption and might require a different model?