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12 min read · updated September 26, 2026

Parlays, and why the cut compounds

A parlay multiplies the payouts, and it multiplies the chances, and it multiplies the cut too. How a parlay pays, how to work out its real chance of hitting, why same-game legs break the math, and what a parlay returns per dollar on average.

The short version

A parlay ties several bets into one ticket. Every leg has to win, and if they all do, the payouts multiply together, which is why a few modest favorites can add up to a big number. But the chance of hitting multiplies too, and so does the cut the sportsbook takes on each leg. On a single bet at the classic −110 price on a true coin flip, you lose about 4.5 cents per dollar on average. Put two of those in a parlay and it's about 9 cents; five and it's about 21. The payout grows fast, but the share the book keeps grows faster. That's the whole idea. The rest of this article is the arithmetic behind it, the one situation where the arithmetic breaks, and how the site's parlay grader checks a ticket for you.

First, one price as a probability

Every price is a probability in disguise. The article on how betting odds work covers this in full; here is the part a parlay needs.

A price of −N implies a chance of N / (N + 100). A price of +N implies 100 / (N + 100). So −150 implies 150 / 250 = 60%, and +100 implies 100 / 200 = 50%.

For parlays, the easiest form of a price is decimal odds: how much comes back for each dollar staked, stake included. Decimal odds are just one divided by the implied probability:

decimal odds = 1 / implied probability

So −150 is 1 / 0.60 = 1.667 (a winning dollar comes back as $1.67), and +100 is 1 / 0.50 = 2.0 (a winning dollar comes back as $2). Going the other way, a decimal price of d "charges" 1 / d as a probability. Hold on to that word, charges. A price is what the book is charging you in probability, and the question is always whether the true chance is higher or lower than the charge.

One more piece. At −110 on both sides, the classic line, each side implies 110 / 210 = 52.4%. Together they add to 104.8%, though the game can only happen 100% of the time. The extra 4.8% is the cut, the book's margin. You'll also hear it called the vig, the juice or the hold; they all mean the same thing. It's why a bet at −110 on a true 50% game isn't a fair bet, and it's the thread that runs through everything below.

How a parlay pays

A parlay's payout is the product of its legs' decimal odds:

parlay payout = leg 1 decimal × leg 2 decimal × leg 3 decimal × ...

The logic is a chain. If the first leg wins, your dollar becomes $1.667. That whole $1.667 then rides on the second leg at 2.0, and becomes $3.333. Each leg bets everything the previous legs won.

So a parlay of −150 and +100 pays 1.667 × 2.0 = 3.333: a $10 ticket returns $33.33 if both legs win. In American odds that's +233 (for decimal odds above 2, the American price is 100 × (decimal − 1), and 100 × 2.333 = 233).

Some books quote their own parlay price instead of the straight product, especially for tickets with many legs or legs in the same game. When a book quotes one, that quoted price is the payout, and the same arithmetic applies to it.

The true chance: multiply, but only for separate games

The chance that every leg wins is the product of each leg's chance:

chance all legs win = leg 1 chance × leg 2 chance × leg 3 chance × ...

If one team has a 60% chance and another has a 50% chance, and the two games have nothing to do with each other, both win 0.60 × 0.50 = 30% of the time. Picture a hundred Sundays: the first team wins on 60 of them, and on half of those 60, the second team wins too. That's 30.

That word independent is doing real work. Multiplying chances is only correct when one leg's result tells you nothing about another's. Two games on different fields, with different teams, are close enough to independent.

Legs from the same game are not. Take the extreme case: a parlay of the home team to win and the away team to win, in the same game, at 60% and 40%. Multiplying says 0.60 × 0.40 = 24%. The real chance is zero, because both can't happen. Real same-game parlays are subtler, but the problem is the same in the other direction too. A team winning and its quarterback having a big day tend to go together, so the chance of both is higher than the product says. Once legs are tied together, the product of their chances is simply the wrong number, and there's no clean way to fix it from the outside. That's why the site's grader takes one leg per game: it only does math it can stand behind.

Expected return: the one number that matters

A parlay has a payout and a true chance. Put them together and you get the expected return per dollar, what the ticket gives back on average, over many tries, for each dollar you stake:

expected return per dollar = true chance × decimal payout − 1

Zero means a fair bet: over the long run you get your money back. Positive means the price is better than the chance. Negative means the book is keeping that much of every dollar, on average.

A fair parlay

Start with the parlay above, and suppose both prices are exactly fair: the −150 team really is 60% and the +100 team really is 50%.

• Payout: 1.667 × 2.0 = 3.333
• True chance: 0.60 × 0.50 = 30%
• Expected return: 0.30 × 3.333 − 1 = 1.000 − 1 = 0

Exactly zero. With no cut on the legs, a parlay of independent games is a fair bet, the same as the singles. The multiplication is honest: a longer payout for a smaller chance, in exact proportion. So far there's no special harm in a parlay.

The same parlay, with the cut

Now price it the way a book actually would. Same teams, same true chances of 60% and 50%, but the favorite is −170 and the coin flip is −110.

• −170 is 1 + 100/170 = 1.588 in decimal (it charges 170 / 270 = 63.0%, against a true 60%).
• −110 is 1 + 100/110 = 1.909 (it charges 52.4%, against a true 50%).
• Payout: 1.588 × 1.909 = 3.032, which charges 1 / 3.032 = 33.0%.
• True chance: still 0.60 × 0.50 = 30%.
• Expected return: 0.30 × 3.032 − 1 = 0.910 − 1 = about −9 cents per dollar.

Now compare with betting each leg alone:

• The −170 single: 0.60 × 1.588 − 1 = −4.7 cents per dollar.
• The −110 single: 0.50 × 1.909 − 1 = −4.5 cents per dollar.

Each single costs a little under a nickel per dollar. The parlay costs about nine. It didn't add the two cuts; it multiplied them.

Why the cut compounds

Here's the cleanest way to see it. On any single, the expected return has a simpler form: the true chance divided by what the price charges, minus one. At −110 on a true 50% game, that's 50 / 52.4 = 0.955. You keep about 95.5 cents of each dollar on average.

A parlay runs the whole stake through every leg in turn, so the keep-rates multiply:

parlay expected return = (keep-rate of leg 1 × keep-rate of leg 2 × ...) − 1

With every leg at −110 on a true coin flip, each leg keeps 0.955, and n legs keep 0.955 to the nth power. (Check it against the long way for two legs: 0.25 × 1.909 × 1.909 = 0.25 × 3.645 = 0.911, which is 0.955 × 0.955. Same answer.)

The payouts come from 1.909 multiplied by itself (1.909, 3.645, 6.958, 13.28, 25.36). The fair column is what the ticket would pay with no cut at all, 1 divided by the true chance. The returns are 0.955 to the first through fifth power, minus one: 0.955, 0.911, 0.870, 0.830, 0.792.

Read the last row slowly. A five-leg ticket that should pay 32 to 1 pays about 25. The number on the ticket, about +2436 in American odds, looks enormous, and it is, but a fair price would be +3100. The gap between them is the compounded cut, and at five legs it's about a fifth of every dollar staked.

This is the honest core of the parlay. It isn't that the math is hidden or crooked. The cut on each leg is the same one you'd pay on a single. It's that stacking legs stacks the cut, and a big payout makes a big cut easy to miss. Most bettors lose over time, and the cut is the main reason; a parlay just applies it more times per ticket.

When a parlay isn't worse

The same arithmetic works in the other direction, which is worth understanding even if it's rarer than it looks.

The multiplication doesn't care whether a leg's keep-rate is above or below 1. Suppose you've found a price that charges less than the true chance. The price check example: DET at +150 charges 1 / 2.5 = 40.0%. If the true chance is 45%, that single's expected return is 0.45 × 2.5 − 1 = +12.5 cents per dollar, a keep-rate of 45 / 40 = 1.125.

Now add a −110 coin flip to it as a second leg:

• Payout: 2.5 × 1.909 = 4.773
• True chance: 0.45 × 0.50 = 22.5%
• Expected return: 0.225 × 4.773 − 1 = 1.074 − 1 = +7.4 cents per dollar

That's 1.125 × 0.955, as the keep-rate rule says. The good leg carried the ticket, but the ordinary leg took about two-fifths of its margin (12.5 cents down to 7.4). Every leg that isn't itself a good price drags the whole ticket toward the book. That's the only situation where a parlay's math holds up: every leg is fairly priced or better, and the games are separate. And all of it rests on the true chances being right, which is the hard part. A forecast of 45% is an estimate, not a fact.

None of this is advice. It's how the prices work. If you do bet, bet only where it's legal, only if you're of age, and only with money you can afford to lose.

The site's parlay grader

The price check has a parlay mode that does all of the above for you. You enter each leg (a game, a side and its price, one leg per game) and, if the book quoted its own parlay price, that too. The grader:

1. Multiplies the legs' decimal odds to get the payout, or uses the book's quoted price if you gave one, and shows what that payout charges (1 divided by the payout).
2. Takes the site's fair number for each leg, the same number the price check uses for singles, and multiplies them for the true chance of the whole ticket.
3. Works out the expected return per dollar: true chance × payout − 1.
4. Gives a verdict: value at +4 cents per dollar or better, bad at −4 cents or worse, fair in between.
5. Marks each leg value, fair or bad, exactly as it would grade that leg as a single, so you can see which leg is costing you.

Run the cut example through it: at −9 cents per dollar, a two-leg ticket of ordinary −170 and −110 prices lands in bad, even though each leg on its own is only a little under a nickel a dollar. Two ordinary legs are enough to cross the line.

If you record parlays in the bet tracker, it grades them after the fact as well, with closing-line value for the whole ticket: the product of each leg's closing fair chance, divided by what the payout charged, minus one. That's the same keep-rate idea, measured against where the market finished instead of against a forecast. The article on closing-line value explains why that's a useful early signal.

Where to see it on the site

• Price check: grade a single or a parlay, see the best price across every book and both exchanges, and flip prices between odds and percents.
• Bets: track singles and parlays, with results, closing-line value and breakdowns by bet type.
• Markets: each game's home win chance from the books, Kalshi and Polymarket side by side.
• How it works: how the site turns prices into probabilities and removes the cut.
• How betting odds work: the full story of a single price.

A few terms

• Leg: one bet inside a parlay. Every leg must win for the parlay to pay.
• Decimal odds: what comes back per dollar staked, stake included. One divided by the implied probability.
• Charges: the probability a price implies, cut included. A payout of 3.032 charges 33.0%.
• The cut: the book's margin, also called the vig, the juice or the hold. At −110 on both sides, 4.8%.
• Independent: two legs whose results don't affect each other. Only then does multiplying chances give the true chance.
• Correlated: legs that tend to win or lose together, like two bets on the same game. Multiplying their chances gives the wrong answer.
• Expected return: the average result per dollar over many tries. True chance × decimal payout − 1.
• Keep-rate: true chance divided by what the price charges. Below 1, the book keeps the difference; a parlay multiplies them.