Same-Game Parlays: The Mathematics of Correlation

Traditional Parlay Mathematics (Independent Events)

For traditional parlays with independent events, the mathematics is straightforward. If you have n bets with individual probabilities p₁, p₂, ..., pₙ, the probability of all bets winning is simply the product of the individual probabilities:

This formula relies on the fundamental rule of probability for independent events. Two events A and B are independent if knowing that A occurred gives no information about whether B occurred (formally: P(A ∩ B) = P(A) × P(B)).

Example: Traditional 3-Leg Parlay

Suppose you parlay three bets from different games (ensuring independence):

• Team A spread -110 (implied probability ≈ 52.4%)
• Team B spread -110 (implied probability ≈ 52.4%)
• Team C spread -110 (implied probability ≈ 52.4%)

Note on implied probability: We convert American odds to probability using the formula from Article 1. For -110 odds:

Combined probability (assuming independence):

Fair odds calculation: If the true probability is 14.4%, the fair payout odds are calculated as:

Actual payout: Most sportsbooks pay approximately 6-to-1 (+600) for a 3-leg parlay at standard -110 odds.

The sportsbook's edge comes from paying slightly less than fair odds. In this case:

This modest house edge is typical for traditional parlays. However, this calculation critically assumes independence—that each outcome does not affect the others. For same-game parlays, this assumption completely breaks down.

The Correlation Problem in Same-Game Parlays

When all bets come from the same game, independence is violated. Consider this common SGP construction:

• Team A to win (-140, implied probability ≈ 58.3%)
• Team A's quarterback to throw for over 275.5 yards (-110, implied probability ≈ 52.4%)
• Game total to go over 48.5 points (-110, implied probability ≈ 52.4%)

These outcomes are positively correlated:

• If Team A wins, their quarterback likely performed well → positive correlation between legs 1 and 2
• If the quarterback threw for 275+ yards, the game likely featured more scoring → positive correlation between legs 2 and 3
• If Team A wins, particularly by a comfortable margin, the total is more likely to exceed expectations → positive correlation between legs 1 and 3

Using the independence formula would drastically underestimate the true probability of all three hitting together.

Mathematical Framework

Let X₁, X₂, X₃ be binary random variables (1 = win, 0 = lose) representing each leg of the parlay. We need to calculate:

Under independence:

With correlation:

The true probability depends on the joint probability distribution of (X₁, X₂, X₃), which captures how the outcomes move together. We cannot simply multiply marginal probabilities; we must account for the dependency structure.

To properly price this parlay, sportsbooks must estimate P(X₁=1, X₂=1, X₃=1) directly, accounting for correlation. The remainder of this article explores the methods they use.

Correlation Matrices: Measuring Dependence

Sportsbooks estimate correlation using historical data. For every pair of bet types (team win with team total, quarterback yards with game total, etc.), they calculate empirical correlation coefficients from thousands of past games.

Pearson Correlation Coefficient

For two binary outcomes X and Y (coded as 1 for win, 0 for loss), the Pearson correlation coefficient is:

The numerator measures how much the joint probability differs from what independence would predict. The denominator normalizes this difference to produce a value between -1 and +1.

Interpretation:

• ρ = +1: Perfect positive correlation (both always happen together)
• ρ = 0: No correlation (independent events)
• ρ = -1: Perfect negative correlation (when one happens, the other never does)
• Typical sports betting range: ρ between -0.4 and +0.6

Example Correlation Matrix

Here is a hypothetical correlation matrix derived from historical data for NFL games. These values are illustrative but representative of the correlations sportsbooks would observe:

This matrix shows moderate positive correlations. The strongest correlation (0.42) is between the quarterback throwing for 275+ yards and the game total going over—this makes intuitive sense, as high passing yardage typically indicates a high-scoring game.

The team winning is positively correlated with both their quarterback's performance (0.35) and the game going over (0.28), though these correlations are weaker. This structure is typical: correlations exist but are rarely extreme in either direction.

Important note: Correlation matrices vary significantly based on game context (favorite vs. underdog, home vs. away, high-total vs. low-total games, etc.). Sophisticated sportsbooks maintain separate matrices for different game situations.

Gaussian Copula Method for Pricing SGPs

One sophisticated approach sportsbooks use is the Gaussian copula, which models joint probabilities while preserving the marginal probabilities of each individual bet. This method separates the marginal behavior (how often each bet wins individually) from the dependence structure (how the bets move together).

The Methodology

1. Transform to normal variables: Convert each binary outcome to a latent continuous normal variable using the inverse normal cumulative distribution function (CDF):
   Z₁ = Φ⁻¹(p₁), Z₂ = Φ⁻¹(p₂), Z₃ = Φ⁻¹(p₃)

   where Φ⁻¹ is the inverse standard normal CDF and p₁, p₂, p₃ are the marginal probabilities.

2. Apply correlation structure: Model (Z₁, Z₂, Z₃) as a multivariate normal distribution with correlation matrix R:
   (Z₁, Z₂, Z₃) ~ MVN(0, R)

   where R contains the pairwise correlation coefficients from the correlation matrix.

3. Calculate joint probability: The probability that all three bets win is:
   P(all win) = P(Z₁ > c₁, Z₂ > c₂, Z₃ > c₃)

   where c₁, c₂, c₃ are the critical values corresponding to each bet not winning (i.e., cᵢ = Φ⁻¹(1 - pᵢ)).

This integral over the multivariate normal distribution is typically computed using Monte Carlo simulation or numerical integration methods.

Worked Example

Using the three-leg parlay from earlier with the correlation matrix shown above:

• P(Team A wins) = 0.583
• P(QB over 275 yards) = 0.524
• P(Total over 48.5) = 0.524

Under independence:

With correlation (using Gaussian copula with the matrix above):

The correlation increases the joint probability by approximately 33% compared to the independence assumption. This is the critical insight: positive correlation makes the parlay more likely to hit than independence would suggest, which means the sportsbook must offer shorter odds (lower payout) than a traditional parlay would provide.

If the book paid traditional 3-leg parlay odds (around +600) when the true probability is 21.2%, they would be offering:

This would be disastrous for the sportsbook. (For a review of expected value calculations, see Article 2.) Instead, they might offer +350, which gives:

The bettor now faces a 4.6% house edge, similar to a single bet.

Empirical Frequency Method

A simpler, more direct approach is to count how often specific bet combinations hit in historical data. This method requires no assumptions about the form of the correlation (unlike the Gaussian copula) and simply uses observed frequencies.

The Process

1. Identify comparable historical games: Find all past games matching the current scenario (similar point spreads, similar totals, similar team strengths)
2. Record outcomes: For each historical game, record whether each bet leg would have won
3. Calculate joint frequency: Count how often all legs hit together
4. Adjust for sample size: Apply statistical adjustments (such as confidence intervals) to account for limited data
5. Add house edge: Convert the frequency to odds with the desired profit margin built in

Example Calculation

From 500 historical NFL games where a team was favored by 3-7 points with a game total between 45-51 points:

Comparison to independence:

This empirical approach confirms what the Gaussian copula predicted: correlation increases the joint probability substantially. The sportsbook uses this 20.4% figure (possibly with adjustments for current game specifics) to set their odds.

Advantages of empirical method:

• No distributional assumptions required
• Captures real-world correlations exactly as they occur
• Easy to implement with sufficient historical data

Disadvantages:

• Requires large datasets for each specific combination
• Doesn't generalize well to novel combinations
• Can be noisy for rare bet types or unusual game contexts

Most sophisticated sportsbooks use a hybrid approach: empirical frequencies where data is abundant, Gaussian copulas or other models to fill in gaps and smooth estimates.

How Sportsbooks Calculate SGP Odds: Complete Process

Here is the complete workflow a sportsbook uses to price a same-game parlay:

Step 1: Estimate Marginal Probabilities

For each individual leg, determine the true probability (before vigorish). Sportsbooks derive these from their predictive models and market-making algorithms:

• Team A to win: 56% true probability → offered at -130 (implied 56.5% with vig)
• QB over 275 yards: 48% true probability → offered at -110 (implied 52.4% with vig)
• Total over 48.5: 52% true probability → offered at -110 (implied 52.4% with vig)

Step 2: Apply Correlation Adjustment

Using either copula methods or empirical frequencies (or both), calculate the true joint probability. For this example, suppose their analysis yields:

Compare to independence assumption:

Correlation increases the joint probability by 35% in this case.

Step 3: Add Vigorish

Convert true probability to offered odds with the desired house edge built in:

This 14.9% house edge is substantially higher than the typical 4-5% edge on a single bet. This is one reason sportsbooks are eager to promote SGPs.

Step 4: Round to Standard Parlay Payouts

Many sportsbooks round to standard parlay increments (+300, +350, +400, +450, +500, etc.) for operational simplicity and better user experience. This rounding can slightly increase or decrease the effective house edge depending on which direction the rounding goes.

In this case, +350 is already a standard increment, so no additional rounding is needed.

Step 5: Dynamic Adjustments

Sophisticated books also make real-time adjustments based on:

• Action imbalance: If too many bettors are taking a specific combination, odds may be shortened further
• Sharp money indicators: If known sharp bettors are avoiding certain SGPs, the book might offer slightly better odds to attract more action
• Correlation uncertainty: For unusual combinations where correlation is hard to estimate, books often add extra margin for safety

Practical Implications for Bettors

Understanding the mathematics of SGPs leads to several practical conclusions:

1. SGPs Are Generally Poor Value

The house edge on SGPs is typically 3-5x higher than single bets. Unless you have strong reason to believe a specific SGP is mispriced, you're better off making individual bets or avoiding parlays entirely.

2. Avoid Highly Correlated Combinations

The combinations that "feel" smartest (Team Win + QB Over + Game Over) are exactly where sportsbooks have the most data and the best pricing models. You're unlikely to find value here.

3. Consider Negative Correlation

If you must bet SGPs, look for negatively correlated combinations where the book may offer disproportionately high payouts. These feel counterintuitive but may offer better mathematical value.

4. Sample Size Requirements

To build your own correlation estimates, you'd need hundreds or thousands of relevant historical games. For most bettors, this is impractical. Recognize that the sportsbook has vastly superior data and modeling capabilities.

5. Alternative Strategy: Single Bets

If you believe Team A will win AND their QB will go over yards AND the game will go over the total, you have three positive EV bets (in your opinion). Why combine them into an SGP with 15-25% house edge when you could make three separate bets with 4-5% edge each? (We discuss optimal sizing for multiple bets in Article 3.)

The SGP costs you roughly 7x more in expected value than the three individual bets, even though both strategies risk the same $10 per combination. This assumes your probability estimates are correct, which brings us back to Article 2 on expected value calculations.