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A parlay is not a long shot. It is the same tax, charged again for every leg.

Long odds are free if the price is fair. What makes a parlay expensive is that the house edge on each leg multiplies, and almost nobody who sells one says so.

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The short version, in dollars

Every figure below assumes the standard −110 price you see on both sides of a point spread, and $100 at risk. The only thing that changes between the rows is how many legs the bet has.

$4.55: what the house expects to keep from $100 on a single bet. This is the number everything else is built from.
$13.03: what it expects from the same $100 on a three-leg parlay. Not three times the single — 2.9 times, because what compounds is the part you keep.
$20.75: at five legs. Split that same $100 across those five bets separately and the expected loss is $4.55.
0.0975: the correlation between two legs that would wipe out the house margin entirely — which is why no book lets you multiply prices on two legs of the same game.

Words we could not avoid

a leg: One of the individual bets inside a parlay. Every leg must win or the whole ticket loses, which is the part everyone already knows.
−110: The standard price on a point spread: risk $110 to win $100. Both sides are priced that way, and the gap between them is the house’s margin.
the hold: The share of everything staked that the book expects to keep. On a single spread bet it is 4.545%. On a parlay it is larger, and this study is about how much larger.
house edge: The same idea as the hold: the fraction of every dollar staked that does not come back on average. Sportsbooks say hold, casinos say edge.
expected value: What a bet is worth on average if you could make it thousands of times. Negative expected value means the average outcome is a loss, however the next ticket lands.
correlation: Whether two outcomes tend to happen together. A quarterback throwing for many yards and his receiver catching many is correlated; two unrelated games are not. Correlation is the whole reason same-game parlays are priced differently.
bankroll: The money you have set aside to bet with — not one stake, but everything you are prepared to lose before you stop.

What this was computed on

Sportsbook pricing is arithmetic, so nothing below is simulated or sampled. The figures follow from published prices and exact rational arithmetic.

The prices

−110 on both sides, the standard point-spread price
Vig removed by scaling the implied probabilities to sum to one
Parlays paid at true multiplied odds, the usual online practice
The common fixed payout chart priced alongside, for comparison

The bets priced

Parlays from one to eight legs
The same stake placed as separate single bets
The payout as a share of a fair one
The correlation between two legs that would erase the margin

The comparison

Equal money at risk, not equal money per leg
Expected loss per $100 staked
Legs assumed independent unless correlation is the subject
No promotions, boosts, free bets or insurance

What is not modelled

Actual same-game parlay pricing, which varies by book and is not published
Odds boosts and promotional pricing, which can genuinely flip a ticket positive
Line shopping across books
Any claim that a bettor can or cannot pick winners — this is about the price, not the picks

We did not measure a single real same-game parlay price. The correlation figure says what a book must charge for; it does not say what any book actually charges, and we have not sampled that. It would need a book’s pricing feed, which is not public.

Everything assumes −110. Books that price at −105 or −115 move every figure here, in the obvious direction.

A promotional boost can genuinely make a parlay positive. That is a marketing decision rather than a property of parlays, and it is why boosts have limits attached.

Finding one

The tax is not added once per leg. It is charged again on what is left.#

single bet: 4.545%
three legs: 13.026%
eight legs: 31.076%
versus the same bets singly: 6.8x

A single spread bet at −110 gives the book 4.545% of the stake. The intuition is that three of them should give three times that, and it does not, because the quantity that survives a leg is what gets multiplied. Keep 95.455% of your expectation three times and you have 86.974% left, which is a hold of 13.026%.

By eight legs the book holds 31.076%. Those same eight bets placed one at a time still hold 4.545%. Nothing about the picks changed; only the packaging did.

This is the exact mirror of the craps study on this site. There, a bet with zero edge could be stacked indefinitely without changing the expected loss. Here every leg carries a real edge, so stacking compounds it. Same arithmetic, opposite sign.

Red is the parlay. Blue is the identical money placed as separate singles. The gap between them is what the packaging costs.

Finding two

A winning parlay is paid less than it is worth, and the shortfall grows.#

one leg pays: 95.5% of fair
three legs: 87.0% of fair
eight legs: 68.9% of fair
three-leg breakeven: 14.37%

Three coin flips are worth 7 to 1. A three-leg parlay at −110 pays 5.96 to 1 — 87.0% of that. At eight legs you are paid 68.9% of fair, so nearly a third of the win has quietly gone before the ticket is even placed.

Put the other way: a three-leg ticket needs all three legs to land 14.37% of the time to break even, against the 52.38% a single bet needs. Those are the same picks. It is the bar that moved.

This is the part that is genuinely hidden. A bettor can see the odds are long. They cannot see that the long odds are short of the true long odds, because nothing on the ticket says what fair would have been.

The blue portion is what you are paid; the dark portion is the part of a fair payout that is not there. It grows with every leg you add.

Finding three

Same-game parlays are priced differently because a tiny correlation would break them.#

independent joint probability: 0.2500
fair would need: 0.2744
required correlation: 0.0975
what books do about it: price it separately

Two legs at even money, priced by multiplying, pay 3.6446. For that to be a fair price both legs must land 27.44% of the time. Independent legs land together 25% of the time, so the ticket is short. But legs from the same game are not independent, and the joint probability rises with their correlation.

Solve for the correlation that closes the gap and it is 0.0975. Under a tenth. That is a small enough relationship to spot by watching a single match — a quarterback and his own receiver, a team total and that team’s star — and it would be enough to erase the entire margin.

Which is exactly why no book multiplies prices on legs from one game. The correlation the bettor believes they are exploiting is the reason the price is different in the first place. We have not measured how much any particular book charges for it; that would need their pricing feed. What the arithmetic shows is that they cannot afford to charge nothing.

Finding four

Two books can offer the same parlay at prices that are not close.#

four legs, true odds: 16.979%
four legs, fixed chart: 31.250%
three legs, which wins: the fixed chart
six legs, fixed chart: 35.938%

Some books pay parlays at true multiplied odds. Others still pay a fixed chart — 2 teams 2.6 to 1, 3 teams 6 to 1, 4 teams 10 to 1, and so on. They are not the same bet. At four legs the chart holds 31.250% against 16.979% for true odds, which is close to double.

The surprise is at three legs, where the old chart is the better of the two: 12.500% against 13.026%. Six to one on three legs is slightly generous relative to multiplying three −110s together. It is the only size where that is true, and it reverses immediately afterwards — by six legs the chart holds 35.938%.

Which means the useful habit is not "avoid parlays" but "read the payout table". The difference between two books on the same four-leg ticket is larger than the entire edge on any single bet either of them offers.

Every parlay size, priced exactly

At −110 a leg, paid at true multiplied odds. The last column is the same money placed as separate single bets, which is the choice actually on offer.

Single bet	4.545%
2-leg parlay	8.884%
3-leg parlay	13.026%
4-leg parlay	16.979%
5-leg parlay	20.753%
6-leg parlay	24.355%
7-leg parlay	27.793%
8-leg parlay	31.076%

What to do with a parlay

There is one honest reason to bet a parlay and several bad ones. Knowing which you are doing is most of the value here.

research/parlays/engine.py · exact arithmetic from published prices

Before you build one

Two questions that decide everything after.

cost of the mistake

Why you are betting it

Most people — A small stake can win a large amount, so the value is in the upside.

Correct — Bet it for the ride, not the return. The upside is real and it is priced.

A three-leg ticket costs 13.026% against 4.545% for the same money placed as singles. You are paying 2.9× for the shape of the payout, and the shape is the only thing you are getting.

$13.03per $100, three legs

not a cost — nothing to compare by length

Whether the legs are from the same game

Most people — Correlated legs are an edge — if the quarterback throws well, his receiver catches well.

Correct — The book already knows. Same-game parlays are priced separately for exactly this reason.

A correlation of only 0.0975 between two legs would erase the entire margin on a product-priced ticket. That is a small enough number to find by watching one match, which is why you are never offered that price.

0.0975correlation that would break even

If you are betting one anyway

The size of the ticket is the whole cost. Nothing else you choose matters as much.

cost of the mistake

How many legs to put on it

Most people — One more leg barely changes anything and doubles the payout.

Correct — Fewer. Every leg multiplies the tax rather than adding to it.

Two legs 8.884%. Five legs 20.753%. Eight legs 31.076% — nearly seven times what those eight bets cost placed one at a time.

$31.08per $100, eight legs

not a cost — nothing to compare by length

An odds boost or a profit boost on the ticket

Most people — A promotion is marketing, so it cannot really change the maths.

Correct — It can, and it is the only thing on this page that does. Read the cap.

A boost large enough to cover the hold genuinely makes a ticket positive — which is why boosts come with maximum stakes attached. The cap is the book telling you exactly how much positive expectation it is prepared to sell, and it is usually 13.026% of a small number.

the capis the part worth reading

cost of the mistake

Whether the book pays true odds or a fixed chart

Most people — A parlay is a parlay.

Correct — Check. At four legs and up the difference is larger than the bet itself is worth.

At four legs, true multiplied odds hold 16.979% and the common fixed chart holds 31.250%. At three legs the old chart is actually the better of the two, at 12.500% against 13.026%.

$14.27per $100, four legs, for reading the wrong chart

The hold compounds. One leg 4.545%, three legs 13.026%, eight legs 31.076%. The same bets placed singly hold 4.545% however many of them there are.

A winning eight-leg parlay pays 69% of fair. Nothing on the ticket tells you what fair would have been.

A correlation of 0.0975 would erase the margin on a two-leg ticket, which is the entire reason same-game parlays carry their own prices.

There is one good reason to bet a parlay: you want a small chance of a large number, and you have decided what that is worth to you. That is a legitimate thing to buy. Just buy it knowing the price, which is not the long odds — it is that the long odds are short.