Pricing 3-Leg SGPs and the Chain Rule

With three legs, dependence becomes a network. The book isn't using one correlation—they're using a dependence structure. This lesson teaches you how to price 3-leg SGPs with precision.

The Chain Rule for 3 Legs

The most transparent way to price a 3-leg SGP from paired history is the chain rule:

Each conditional term is estimated on the subset of games where the earlier legs already hit.

Why Independence Fails Dramatically for 3 Legs

Let's see how bad the independence assumption gets with more legs:

With 3+ legs, you need all correlations to align favorably. If even one is zero or negative, the SGP becomes extremely unlikely.

Worked Example: NHL 3-Leg SGP (The Trap)

This real-world example shows how "obvious" correlations can be mirages.

The Setup

Carolina Hurricanes same-game parlay:

• Leg A: Seth Jarvis to score a goal (+155)
• Leg B: Andrei Svechnikov to score a goal (+185)
• Leg C: Seth Jarvis Over 1.5 points (+225)
• 3-leg SGP: +900

The Story: If Jarvis scores (1 point for goal), he only needs 1 more point (assist or another goal) to hit 2+ points. And if both forwards score, it suggests a high-scoring game.

Step 1: Convert Odds to Probabilities

Step 2: Check the Game Logs (53 Games)

Step 3: Calculate Independence Baseline

If the three legs were independent:

P(A ∩ B ∩ C) = 0.358 × 0.302 × 0.151 = 0.0163 (1.63%)

Fair odds at independence: +6035

Step 4: Apply the Chain Rule

Let's use the chain rule with actual data:

P(A) = 19/53 = 0.358

P(B|A) = P(B and A) / P(A)

• Games where both Jarvis AND Svechnikov scored: 3/53
• P(B|A) = 3/19 = 0.158

Wait—this is lower than P(B) overall (0.302). There's negative correlation between their goals!

P(C|A ∩ B) = P(all three) / P(A and B)

• All three hit: 1/53
• A and B hit: 3/53
• P(C|A ∩ B) = 1/3 = 0.333

Joint probability:

P(A ∩ B ∩ C) = 0.358 × 0.158 × 0.333 = 0.0189 (1.89%)

This matches our empirical count of 1/53!

Step 5: The Correlation Reality Check

Let's examine the pairwise correlations:

Why the negative A-B correlation?

• When Jarvis scores in a low-scoring game (1-0 or 2-1), Svechnikov often doesn't
• Defensive adjustments after one player scores
• Limited high-scoring games overall

Step 6: Compare to Market

The market is overpricing this SGP by more than 5x!

The One Time It Hit

Out of 53 games, this SGP hit exactly once: January 24, 2026.

This was likely a high-scoring game where everything clicked. But it happened once in 53 games (1.89%). If you bet this SGP every game at +900, you would lose money in the long run.

A Practical r-Based Shortcut for 3 Legs

If you don't have enough paired samples for all conditionals, combine two legs into a synthetic event:

Step 1: Price the First Two Legs

pAB = pA × pB + rAB × √(pA(1-pA) × pB(1-pB))

Step 2: Apply r-Adjustment for Third Leg

pABC = pC × pAB + rC,AB × √(pC(1-pC) × pAB(1-pAB))

This requires two correlation assumptions: rAB and rC,AB.

Choosing the Right Method

Key Lessons for 3-Leg SGPs

Practice Exercise

Context Matters: The Weather Example

Remember the Williams-Loveland SGP from earlier? Let's reveal what actually happened.

The Game: Bears vs. Rams, January 18, 2026

Weather at Soldier Field:

• Temperature: 18°F at kickoff (lows near 4°F)
• Wind chill: As low as -15°F
• Wind: 15-25 mph gusts up to 40 mph
• Precipitation: Consistent light snow, ~1 inch accumulation

The Results

The Lesson

The historical correlation analysis said Williams and Loveland were highly correlated. But we missed context: brutal weather suppresses passing efficiency and TD scoring.

Key Takeaways

1. Chain rule for 3 legs: P(A ∩ B ∩ C) = P(A) × P(B|A) × P(C|A ∩ B)
2. Independence fails dramatically for 3+ legs—errors compound
3. Check pairwise correlations before assuming positive dependence
4. One negative correlation can make a 3-leg SGP nearly impossible
5. "Obvious" stories are often mispriced—the book knows the story too
6. Context (weather, injuries, matchups) can override historical trends