Useful Poisson Properties and Mental Shortcuts
Several Poisson properties are especially useful in prop markets. Memorizing these will let you sanity-check bets quickly—even without opening Excel.
Key Insight
Memorize two shortcuts—Var(X) = λ and P(X=0) = e^(-λ)—and you can sanity-check many markets quickly.
Property 1: Mean Equals Variance
For a Poisson random variable X ~ Poisson(λ), the mean and variance are both λ. That means the standard deviation is √λ.
| λ Value | Mean | Variance | Std Dev (σ) |
|---|---|---|---|
| 1 | 1 | 1 | √1 = 1.00 |
| 4 | 4 | 4 | √4 = 2.00 |
| 9 | 9 | 9 | √9 = 3.00 |
| 0.64 | 0.64 | 0.64 | √0.64 = 0.80 |
Why This Matters for Betting
This gives you a quick sense of uncertainty. Low-λ props have high relative volatility:
- If λ = 1.0, one standard deviation is 1.0 (100% of the mean!)
- If λ = 9.0, one standard deviation is 3.0 (only 33% of the mean)
Warning
Low-λ props like "anytime TD" and "to hit a home run" markets can be noisy even when your model is good. The variance is inherent to the stat, not a flaw in your analysis.
Property 2: The Zero Shortcut
The probability of zero occurrences is always:
Probability of Zero Events
P(X = 0) = e^(-λ)=EXP(-λ)This is incredibly useful for anytime markets (Over/Under 0.5):
| λ | P(X = 0) | P(X ≥ 1) = P(Over 0.5) |
|---|---|---|
| 0.3 | 74.1% | 25.9% |
| 0.5 | 60.7% | 39.3% |
| 0.7 | 49.7% | 50.3% |
| 0.8 | 44.9% | 55.1% |
| 1.0 | 36.8% | 63.2% |
| 1.2 | 30.1% | 69.9% |
| 1.5 | 22.3% | 77.7% |
Quick Mental Math
If λ = 0.6, then P(0) ≈ e^(-0.6) = 54.9%, so the player fails to score more often than not.
Tip
For anytime TD bets, if the player's λ is below ~0.7, there's a better than 50% chance they DON'T score. Many recreational bettors don't realize how often "anytime" props lose.
Property 3: Additivity (When Independence Is Reasonable)
If two independent Poisson processes have rates λ₁ and λ₂, then their sum is Poisson with rate λ₁ + λ₂.
Practical Use: "At Least One Happens"
Example: Two receivers on the same team. Receiver A has λ₁ = 0.8 TDs. Receiver B has λ₂ = 0.6 TDs. What's P(at least one of them scores)?
Method: Use the complement (neither scores).
P(at least one) = 1 − P(both are 0)
= 1 − e^(-0.8) × e^(-0.6)
= 1 − e^(-1.4)
≈ 1 − 0.247
= 75.3%
Note
This assumes independence, which isn't perfectly true (they're competing for the same TDs). But it's a useful approximation for quick analysis.
Property 4: Normal Approximation at High λ
When λ is large (rule of thumb: λ > 20), Poisson is well-approximated by a normal distribution with:
- μ = λ
- σ = √λ
This is one reason Chapter 7's normal distribution works well for high-volume stats.
When to Switch
| λ Range | Best Approach |
|---|---|
| 0-5 | Poisson (discrete, skewed) |
| 5-20 | Poisson (still best, but normal okay for rough estimates) |
| 20+ | Normal approximation (μ = λ, σ = √λ) |
Example: A stat with λ = 25
- Poisson: Calculate exact probabilities
- Normal approximation: μ = 25, σ = 5
- Use NORM.DIST for quick estimates
Multiple-Outcome Markets: Exact Counts and Tail Pricing
Poisson is also useful for markets that pay different prices for different exact outcomes, such as "exactly 2 passing TDs" or "4+ passing TDs."
Example: Patrick Mahomes Passing TDs (λ = 2.3)
| Outcome | Excel Formula | Probability |
|---|---|---|
| P(0) | =POISSON.DIST(0, 2.3, FALSE) | 10.0% |
| P(1) | =POISSON.DIST(1, 2.3, FALSE) | 23.1% |
| P(2) | =POISSON.DIST(2, 2.3, FALSE) | 26.5% |
| P(3) | =POISSON.DIST(3, 2.3, FALSE) | 20.3% |
| P(4) | =POISSON.DIST(4, 2.3, FALSE) | 11.7% |
| P(5+) | =1-POISSON.DIST(4, 2.3, TRUE) | 8.4% |
Where the Edge Lives
Key Insight
A common edge in these markets is the tail: books often underprice extreme outcomes (like 4+ TDs) even when they price the "most likely" counts correctly.
Books focus on getting the middle outcomes right (1-2 TDs) because that's where most action lands. The tails (0 TDs and 4+ TDs) are often mispriced.
Quick Reference: Poisson Mental Shortcuts
| Shortcut | Formula | Use Case |
|---|---|---|
| P(zero) | e^(-λ) | Anytime props, Under 0.5 |
| Variance = Mean | Var(X) = λ | Assess volatility |
| Std Dev | σ = √λ | Confidence intervals |
| Combined rate | λ₁ + λ₂ | Multiple independent events |
| Normal approx | μ = λ, σ = √λ | When λ > 20 |
📝 Exercise
Instructions
Test your mastery of Poisson properties with these quick calculations.
A player has λ = 0.5 touchdowns per game. Using the zero shortcut, what's the approximate probability they DON'T score (P(X=0))?
Two independent events have λ₁ = 0.9 and λ₂ = 0.5. What's the approximate probability that at least one occurs?
A prop has λ = 4. What's the standard deviation of outcomes?
You're analyzing a stat with λ = 35. Which approach is most appropriate?