Correlation

The independence assumption — and why it fails

Beginner parlay math goes like this: leg 1 wins 50% of the time, leg 2 wins 50% of the time, so the parlay wins 25% of the time. Multiply the probabilities, done. This works only when the two legs are independent — when knowing the outcome of leg 1 tells you nothing about leg 2.

Independence is rare in sports betting. Most natural parlay combinations involve game-script dependence: if the offense moves the ball, multiple offensive props correlate. If a team blows out the opponent, spread, total, and ML all correlate. If a pitcher gets shelled, his strikeout prop and the team's run total correlate. Books know this; they price for it. The customer often doesn't.

Pearson correlation — the basic measure

# Pearson correlation coefficient r
r = cov(X, Y) / (sd(X) × sd(Y))

# Range: -1 (perfectly negative) to +1 (perfectly positive)

# In sports betting context
r = +1.0: identical bets (e.g., Lakers ML and Lakers ML)
r = +0.7: strong positive (QB passing yards Over + WR1 receiving yards Over)
r = +0.4: moderate positive (Team total Over + game total Over)
r =  0.0: independent (Lakers ML + Cowboys ML, different sports)
r = -0.3: moderate negative (Lakers ML + Lakers spread covered)
r = -1.0: opposing (Lakers win + Celtics win on same game)

How correlation breaks parlay math

Conditional probability formula: P(A and B) = P(A) × P(B|A)

If A and B are independent: P(B|A) = P(B), and joint = P(A) × P(B).

If A and B are positively correlated: P(B|A) > P(B), and joint > P(A) × P(B).

If A and B are negatively correlated: P(B|A) < P(B), and joint < P(A) × P(B).

# Example: QB Over 250 yards + Team Total Over 24.5
P(QB Over)             = 0.50
P(Team Over)           = 0.50
P(Team Over | QB Over) = 0.85   # conditional probability

# Joint probability (true)
joint = 0.50 × 0.85 = 0.425  # 42.5%

# Naive independent calculation
naive = 0.50 × 0.50 = 0.25   # 25%

# Difference = 17.5 percentage points
# True parlay should pay ~ +135 (decimal 2.35)
# Naive math says +300 (decimal 4.00)
# Book typically prices ~+150 to +200 — capturing correlation

SGP pricing — books embed correlation

Modern US sportsbooks (DraftKings, FanDuel, BetMGM, Caesars) all run Monte Carlo simulation pricers for same-game parlays. Process:

1. Model the underlying game with statistical distributions (e.g., QB yards ~ N(265, 60), team total ~ N(24, 8)).
2. Apply empirical correlation matrix derived from historical games.
3. Simulate 10,000-100,000 games.
4. Count joint outcomes to estimate joint probability.
5. Convert to price, add vig (typically 4-8% per leg, accumulated).

The customer sees a final price. Behind the scenes, the price reflects the book's best estimate of correlation. If the book's correlation matrix is accurate, the customer gets a fair (vig-adjusted) price. If it's miscalibrated — the model says r = 0.5 but the true value is r = 0.7 — the customer overpays or underpays accordingly.

Where books miscalibrate correlation

Common book weaknesses:

• Game-script-dependent positive correlation — when one team falls behind, they pass more, the QB's volume increases, multiple offensive props correlate. Books often under-price this in trailing-team scenarios.
• Garbage-time corrections — late-game scoring patterns in blowouts vs. close games. Books model average distributions; actual conditional outcomes differ.
• Weather correlation — high wind reduces passing yards, increases rushing, lowers totals. Books adjust each prop individually for weather but often miss the cross-prop correlations.
• Pace correlations — fast-pace teams in basketball produce both more total possessions and more total scoring; both correlate positively. Books model pace but sometimes mis-attribute the correlation strength.
• Star-player dependence — if Tatum scores 40, Celtics likely win, team total likely covers. Books often slightly under-price the star-player-driven correlation.

Negative correlation — the hidden tax

Negatively correlated parlays look great on paper but pay less than expected because joint probability is lower than naive multiplication suggests.

# Lakers ML + Lakers spread doesn't cover
P(Lakers ML)                       = 0.60
P(Lakers spread covered)           = 0.50
P(spread covered | Lakers ML)      = 0.83  # positive if they win
P(spread not covered | Lakers ML)  = 0.17

# Trying to parlay Lakers ML + Lakers DOESN'T cover
P(joint) = P(LAL ML) × P(spread NOT covered | LAL ML)
         = 0.60 × 0.17 = 0.102  # 10.2%, fair odds ~+880

# Naive calculation
P(naive) = 0.60 × 0.50 = 0.30  # 30%, naive odds +233

# Books offering "+450 boost" on this look attractive
# But actual fair price is +880 — bettor is paying massive hidden vig

The recreational bettor sees "+450 boost" and feels they're getting value. The true fair price is +880. The book is collecting roughly 38% effective vig. The bettor doesn't realize because they can't compute the conditional probability.

Correlation in advanced strategies

Correlated parlays as a sharp strategy

Some sharps build parlays of positively correlated legs when their model says the correlation is stronger than the book's. The pitch: book prices QB Over + Team Over at +180 (assuming r = 0.5). Sharp model says r = 0.7, true joint probability = 0.45, fair price = +122. Sharp takes the parlay at +180 with documented edge.

This works only when: ① sharp has correlation data the book doesn't; ② sharp has accurate model of underlying distributions; ③ book's correlation model is materially miscalibrated. Most pros find this strategy works on specific NFL same-game prop combinations and rarely on other sports.

Decorrelation as risk management

For straight bettors running multiple bets on a Sunday, betting decorrelated games (different teams, different game scripts) smooths variance. Industry advice: don't have all your bets on home favorites Week 1 because correlation amplifies a bad week. Pro shops explicitly diversify across game times and game-script types to keep portfolio correlation low.

Empirical correlation in major sports

The "sharp opportunity" cells are where sharp bettors have documented edge betting these correlated combinations as SGPs. They require active correlation modeling and access to historical data. Not for casual bettors.

Common correlation mistakes

• Treating SGP legs as independent — multiplying independent probabilities will overstate parlay probability if positive correlation exists.
• Ignoring negative correlation entirely — bettors miss that some combinations are hidden lottery tickets.
• Assuming book correlation is accurate — books are usually close but not perfect; small miscalibrations create opportunities.
• Single-game bankroll concentration — multiple bets on the same game create high positive correlation in your portfolio.
• Using Pearson r where Spearman or copula is needed — Pearson assumes linear relationships; sports outcomes often have nonlinear dependence (tail correlation in blowouts).

Sources & further reading

• Embrechts, Paul, McNeil, Alexander, Straumann, Daniel. "Correlation and dependence in risk management: properties and pitfalls." Risk Management: Value at Risk and Beyond, 2002.
• Spanos, Aris. "Probability theory and statistical inference." Econometric Theory, 2019.
• Spann, Martin & Skiera, Bernd. "Sports forecasting: a comparison of the forecast accuracy of prediction markets, betting odds and tipsters." Journal of Forecasting, 2009.
• Buchdahl, Joseph. "Hidden correlation in same game parlays." Football Data Blog, 2022.
• DraftKings Engineering Blog — "Pricing same-game parlays with Monte Carlo simulation" (technical methodology).

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