Price the slip, not the legs.

◆ THE MATHS, PLAINLY

Leg probability

Our number, shrunk toward the de-vigged market by weight d / (d + 0.04), where d is the gap between the two. Half weight at a 4pp disagreement, capped at 0.75. A straight bet pays for one estimate, a slip multiplies several, so the far-from-market legs get discounted hardest.

Correlation

A pairwise table by sport, market pair, relationship (same player, same team, opposing sides) and side agreement. Same player receptions over with receiving yards over sits at +0.60. A quarterback's passing yards over with his receiver's receiving yards over at +0.35. A back's rushing yards over with the game total over at +0.25. Flip one side and the sign flips with it.

The joint probability

A one-factor Gaussian copula: each leg becomes a latent normal, one shared factor carries the game script, and the joint is a single numerical integral. Exact for two legs, an approximation past that, with no tail dependence and a known weak case when a slip mixes two separate games. Clamped to the arithmetic bounds so the answer can never be impossible.

Boost, gate and stake

A profit boost multiplies the profit, not the payout: 1 + (dec - 1) x (1 + boost). A max-extra-winnings cap is a dollar ceiling, so it is applied at the recommended stake and the stake is solved for. The gate is the raw price at which the edge falls to +1%. The stake is quarter Kelly on the correlated probability, floored at 0.25u and capped at 1u from three legs up.