Parlays: the math books do not advertise
Parlays are the most heavily promoted product in sports betting. The math is unflattering. A precise look at the implied edge of a multi-leg ticket.
Parlays are the most popular bet type by retail volume and the worst-priced product on the board. The math is not a matter of opinion. Multiplying juice across legs creates an implied book hold that is several times the hold on the underlying singles. There are narrow exceptions; we will get to them.
How parlay payouts are computed
A parlay payout is the product of each leg's decimal odds. If all three legs win, you collect the product. If any leg loses, the entire ticket loses.
THREE-LEG PARLAY OF -110 LEGS each leg decimal: 1.909 parlay multiplier: 1.909 × 1.909 × 1.909 = 6.957 payout on $10 stake = $69.57 returned profit = $59.57 in American = +595
That is the standard book payout for three -110 legs. Now compare it to the fair price.
Fair price vs book price
THREE INDEPENDENT 50% EVENTS (NO JUICE)
prob(all win) = 0.5 × 0.5 × 0.5 = 0.125
fair odds = 1 / 0.125 = 8.000 decimal
= +700 American
BOOK OFFERS
= 6.957 decimal
= +595 American
BOOK HOLD ON THE TICKET
6.957 / 8.000 = 0.870
hold = 1 - 0.870 = 13.0%The book is keeping 13% of the parlay's expected value. That is roughly three times the hold on each individual leg. The hold compounds with each leg added.
| Legs | Fair odds | Book pays | Hold |
|---|---|---|---|
| 2 | +264 | +264 | ~4.8% (matches single leg) |
| 3 | +700 | +595 | ~13% |
| 4 | +1428 | +1228 | ~16% |
| 5 | +2812 | +2434 | ~19% |
| 6 | +5450 | +4795 | ~22% |
| 10 | +98209 | ~+78420 | ~25%+ |
The two-leg parlay is the only one priced in line with single-bet hold. Three legs and up, the hold is brutal.
Parlay Probability
// WB://TOOLS/PARLAY-PROBParlay analysis
Implied probability assumes legs are independent. Same-game parlays are correlated; book SGPs adjust pricing in the book's favor. Fair probability uses your supplied estimates if any leg has one, otherwise the implied probability of the leg's odds. Expected value: +0.00%.
Why parlays exist anyway
Two reasons. First, customers love them. The rare big-payout hit is highly memorable; the much more frequent losses are not. Second, books make a higher margin per ticket on parlays than on singles. Books promote parlays heavily because the unit economics are favorable for the book.
From the bettor's side, the appeal is psychological: a $10 ticket that pays $50 if four longshots all hit is more exciting than four separate small singles. The expected value of those two propositions is, on the math, dramatically different.
Same-game parlays
Same-game parlays (SGPs) combine multiple legs from the same game. They are the most heavily promoted product in modern sports betting and structurally even worse for the bettor than traditional parlays.
The reason is correlation. Legs in an SGP are correlated. A parlay containing 'Mahomes over 287.5 yards' and 'Chiefs win' is not two independent bets, because if Mahomes throws for over 287.5 yards, the Chiefs are more likely to win. Standard parlay math (multiplying odds) overstates the true probability of the joint outcome.
Books know this. They use proprietary correlation models to price the SGP. The pricing accounts for correlation, but in the book's favor: the book's correlation adjustments add hold rather than removing it. A typical SGP runs 15% to 35% hold depending on legs and book.
When parlays can be positive EV
There is a narrow case. If you have positive EV on each leg of a parlay (the bet is +EV as a single), then a parlay of +EV singles is also +EV when the legs are uncorrelated. The math:
TWO INDEPENDENT +5% EV SINGLES EV per leg = +5% parlay EV% ≈ +10.25% (slightly more than additive) But variance is dramatically higher. Bet sizing for the parlay must be much smaller.
In practice, the book's parlay multiplier (and any same-game correlation pricing) usually erases the EV advantage. And the variance amplifies. A bettor with single-bet edge generally compounds bankroll faster by betting the singles, not by parlaying them.
Free parlay tokens, profit boosts, and odds boosts
These are book promotions. Some are genuinely +EV (a free $20 parlay token has positive expected value because there is no downside). Others are decoration on a high-hold product. Read the math each time. A 50% profit boost on a 10-leg parlay still leaves the book with significant positive hold; the boost makes the bet less bad, not necessarily good.
What to read next
Bet types covers the alternatives. Expected value is the math that explains why most parlays are bad bets. How sportsbooks make money explains why books push parlays so hard.