Worked Examples: Touchdowns and Strikeouts
Let's apply the 3-step Poisson process to two real-world prop betting scenarios. These examples will show you exactly how to move from a projection to an expected value calculation.
Worked Example 1: Justin Jefferson Receiving Touchdowns
The Setup
Market Line: Over 0.5 receiving TDs at -150; Under 0.5 at +120
Context:
- Jefferson is averaging 0.85 touchdowns per game over his last 10 starts
- The Bears rank 28th in red zone defense
- The Vikings are 7-point favorites, suggesting multiple scoring opportunities
Step 1: Estimate λ
One reasonable way to build λ:
| Factor | Value | Rationale |
|---|---|---|
| Season average | 0.75 TDs/game | Baseline |
| Matchup adjustment | ×1.15 (+15%) | Bears rank 28th vs WR TDs |
| Game script | ×1.10 (+10%) | Vikings favored by 7 |
| Your λ | 0.95 | 0.75 × 1.15 × 1.10 |
Step 2: Convert to Probability
For an Over 0.5 bet, you need P(X ≥ 1). In Excel:
=1 - POISSON.DIST(0, 0.95, TRUE)
Result: P(Over) = 0.613 (61.3%), so P(Under) = 38.7%
Step 3: Compare to Market
Market implied probability at -150:
150 / (150 + 100) = 60.0%
Your edge: 61.3% − 60.0% = +1.3%
Expected Value Calculation
Risk $150 to win $100:
EV = (0.613 × $100) − (0.387 × $150)
EV = $61.30 − $58.05
EV = +$3.25 per $150 risked
Key Insight
Even a small edge is only as good as your λ estimate—if λ is off, the EV is off. A 1.3% edge is real but requires high confidence in your projection.
Visualizing the Distribution
With λ = 0.95, here's how the probability mass is distributed:
| Outcome | Probability | Cumulative |
|---|---|---|
| 0 TDs | 38.7% | 38.7% (Under) |
| 1 TD | 36.7% | 75.4% |
| 2 TDs | 17.4% | 92.8% |
| 3+ TDs | 7.2% | 100% |
The bar at k=0 (38.7%) is the Under 0.5 outcome. Everything else (61.3%) is the Over 0.5.
Worked Example 2: Gerrit Cole Strikeouts
The Setup
Market Line: Over 7.5 strikeouts at -110; Under 7.5 at -110
Context:
- Cole's season average is 7.2 K/start
- He's on a recent hot streak
- But he's facing the Red Sox, who are harder to strike out
Step 1: Estimate λ
| Factor | Value | Rationale |
|---|---|---|
| Season average | 7.2 K/start | Baseline |
| Recent form | ×1.05 (+5%) | Hot streak |
| Matchup | ×0.90 (−10%) | Red Sox are harder to strike out |
| Your λ | 6.8 | 7.2 × 1.05 × 0.90 |
Step 2: Convert to Probability
Over 7.5 means 8+ strikeouts. In Excel:
=1 - POISSON.DIST(7, 6.8, TRUE)
Result: P(Over 7.5) = 0.372 (37.2%), so P(Under 7.5) = 0.628 (62.8%)
Step 3: Compare to Market
Break-even probability at -110: 52.4%
Your Under edge: 62.8% − 52.4% = +10.4%
This is a substantial edge!
Expected Value Calculation
Risk $110 to win $100 on the Under:
EV = (0.628 × $100) − (0.372 × $110)
EV = $62.80 − $40.92
EV = +$21.88 per $110 risked
Key Insight
When your probability is far from the market's break-even rate, you don't need to be perfect to have value. A 10%+ edge gives you margin for error in your λ estimate.
The Distribution at λ = 6.8
| Strikeouts | P(exactly k) | P(k or fewer) |
|---|---|---|
| 0-4 | — | 16.8% |
| 5 | 12.2% | 29.0% |
| 6 | 13.8% | 42.8% |
| 7 | 13.4% | 56.2% |
| 8 | 11.4% | 67.6% |
| 9 | 8.6% | 76.2% |
| 10+ | — | 100% |
The Under 7.5 includes all outcomes ≤7 strikeouts (62.8% probability). The Over 7.5 is everything ≥8 (37.2%).
Side-by-Side Comparison
| Factor | Jefferson TDs | Cole Strikeouts |
|---|---|---|
| Your λ | 0.95 | 6.8 |
| Line | Over 0.5 | Under 7.5 |
| Your Probability | 61.3% | 62.8% |
| Market Break-even | 60.0% | 52.4% |
| Edge | +1.3% | +10.4% |
| Bet Quality | Marginal | Strong |
Tip
The Cole strikeouts bet has 8× the edge of the Jefferson TD bet. Both might be "correct" projections, but the Cole bet gives you much more margin for error. Prioritize larger edges when allocating your bankroll.
Practice With the Calculator
Try adjusting the λ values in these examples to see how the probabilities change:
Poisson Calculator
Try the interactive calculator for this concept
📝 Exercise
Instructions
Work through these scenarios using the 3-step Poisson process.
A running back has λ = 0.60 rushing TDs. The market offers Over 0.5 TDs at -135. What's your P(Over) and does an edge exist?
A pitcher has λ = 9.2 strikeouts. What's P(Over 8.5) using Poisson?
You calculate +5.2% edge on an Over prop at -120 odds. Is this a bet, a pass, or marginal?