What a parlay price actually is
A parlay is a single bet that pays only if every selection on it wins. Because the outcomes are treated as independent, the price is the product of the individual prices in decimal form, and the probability is the product of the individual probabilities. That multiplication is the entire mechanism, and it is also why parlays feel so generous: three coin flips at even money combine to 2 × 2 × 2 = 8.00, a price that looks nothing like the components.
The trap is that the bookmaker's margin multiplies too. Each leg is priced with a small tax built in, and stacking legs stacks that tax. A single −110 bet asks you to overcome 4.76% of overround. A three-leg parlay at −110 asks you to overcome it three times, which is why the fair value of the standard three-teamer is 12.5% of a payout the ticket prices at 14.37%.
Decimal odds are the natural language here because they already include the stake. A decimal price of 1.91 means every dollar staked returns 1.91 dollars gross. Multiply the decimals and you have the gross return per dollar for the whole ticket, no conversion needed.
The formula, and how the margin is removed
Start by converting every leg to decimal. A positive American price of +150 becomes 1 + 150/100 = 2.50. A negative price of −110 becomes 1 + 100/110 = 1.9091. The combined price is D = d1 × d2 × … × dn, the payout is stake × D, and the profit is that payout minus your stake.
The implied probability of the ticket is simply 1/D. That number is not the probability the ticket wins, because it includes the margin. To get closer to a fair probability you have to strip the margin out of each leg first. If the two sides of a market sum to more than 100% of probability, the excess is the overround. A market priced −110 on both sides implies 52.38% + 52.38% = 104.76%, so the overround is 4.76%. Dividing each side by 1.0476 gives 50.0% and 50.0%, which is the multiplicative de-vig and the method used here.
Applying it leg by leg gives the fair probability of the whole ticket: pfair = ∏(1/dk) / (1 + o). The expected value of the ticket follows immediately as pfair × payout − stake. Every one of those numbers depends on the overround you enter, so treat the fair probability as an estimate whose quality is exactly the quality of that input. If you know the actual price of the other side of each market, compute the real overround from those two prices rather than accepting the default.
Multiplicative de-vig is one of several conventions. Additive de-vig subtracts the same number of probability points from each side, and the power method raises each implied probability to a common exponent. They agree closely near even money and diverge on lopsided markets, where the multiplicative method is known to leave longshots a little overpriced.
Worked example: the standard three-team parlay at −110
Three sides, each priced at −110, on a $100 ticket.
- Convert each leg. −110 → 1 + 100/110 = 1.90909.
- Multiply. 1.90909 × 1.90909 = 3.64463, and 3.64463 × 1.90909 = 6.95793.
- Payout and profit. $100 × 6.95793 = $695.79 back, so the profit is $595.79. Expressed in American odds that is (6.95793 − 1) × 100 = +595.79, which is why most books pay a flat 6-to-1 on a three-teamer and keep the difference.
- Implied probability. 1 ÷ 6.95793 = 0.143721, or 14.372%.
- De-vig each leg. Implied 1/1.90909 = 0.523810 per leg. Divide by 1.047619: 0.523810 / 1.047619 = 0.500000. Each leg is a genuine coin flip once the margin is removed.
- Fair parlay probability. 0.5 × 0.5 × 0.5 = 0.125, or 12.5%.
- Expected value. 0.125 × $695.79 = $86.97 of fair value against a $100 stake, so the expected value is −$13.03 — a hold of 13.0% on the ticket.
Compare that with betting the same three sides individually: each single carries a 4.76% overround, so the expected loss on $100 spread across three singles is about $4.55. Parlaying them roughly triples the cost of the margin without changing a single opinion about the games. The Kelly criterion calculator makes the same point in staking terms: with a negative edge the growth-optimal stake is zero.
How to read the numbers
Compare implied probability against fair probability, and the size of the gap is the house's cut on your ticket. In the worked example the gap between 14.372% and 12.5% is 1.87 percentage points of probability, which becomes 13.0% of the stake once it is expressed as expected value. That percentage grows with every leg you add at the same margin.
The expected value figure is the honest bottom line, and it is negative for essentially every parlay built out of standard sportsbook prices. It turns positive only if you either beat the price on individual legs, or the legs are positively correlated in a way the book has not priced. Correlation is the one genuine parlay edge that exists, and it is also the reason books restrict same-game parlays and price them with their own correlated models rather than by multiplying.
The running-payout column in the table shows where the compounding accelerates. Adding a fourth −110 leg to the three-team example lifts the payout from $695.79 to $1,328.33 while dropping the fair probability from 12.5% to 6.25%. You are paying roughly twice for something that is exactly half as likely, which is the trade a parlay always offers.
Parlay price and fair value at −110 per leg
| Legs | Combined decimal | True American odds | Implied probability | Fair probability | Typical flat book payout |
|---|---|---|---|---|---|
| 2 | 3.6446 | +264.46 | 27.437% | 25.000% | 13 to 5 (+260) |
| 3 | 6.9579 | +595.79 | 14.372% | 12.500% | 6 to 1 (+600) |
| 4 | 13.2833 | +1,228.33 | 7.528% | 6.250% | 10 to 1 (+1000) |
| 5 | 25.3591 | +2,435.91 | 3.943% | 3.125% | 20 to 1 (+2000) |
| 6 | 48.4127 | +4,741.27 | 2.066% | 1.563% | 40 to 1 (+4000) |
| 8 | 176.4464 | +17,544.64 | 0.567% | 0.391% | 150 to 1 (+15000) |
The flat payout schedules are the round numbers commonly posted; they are shown for comparison and vary by operator, so check your own book's parlay card.
Where parlay math goes wrong
- Multiplying correlated legs. A quarterback throwing for 300 yards and his team covering the spread are not independent events. Multiplying their prices understates the true probability, which is exactly why books build separate same-game parlay pricing rather than letting you do it.
- Treating the implied probability as the real one. The 14.37% on a three-teamer includes three layers of margin. The fair number is 12.5%, and the difference is the whole business model.
- Comparing a flat book payout to the true price. Many books pay a fixed 6-to-1 on a three-team −110 parlay, which is +600 against a true +595.79. That one is fractionally in your favour; the four-team flat payout of 10-to-1 against a true +1228 is not.
- Assuming a push cancels the ticket. On most parlay cards a pushed leg is removed and the ticket reprices as though it had one fewer leg. A few operators grade it as a loss. Read the house rules before you assume.
- Adding legs to chase a payout. Each extra leg multiplies the payout by roughly two but also multiplies your probability of losing everything. The expected value moves against you every time.
- Using the same overround for every market. Point spreads run near 4-5%, while long futures markets can carry 20% or more. Enter the real number for the market you are betting.
Parlays, teasers and round-robins
A teaser is a parlay in which every leg is moved a fixed number of points in your favour in exchange for a shorter price. The arithmetic is identical — multiply the teased prices — but the probabilities change because the lines have moved, so you cannot reuse the original leg probabilities. A round-robin is not one bet at all: it is a set of smaller parlays covering every combination of the selections, so its cost is the number of combinations multiplied by the unit stake, and each sub-parlay prices exactly as this calculator prices it.
If you want to know how much to stake rather than what a ticket pays, the growth-optimal answer comes from the Kelly criterion bet size calculator, using the combined price and the combined probability from this page. If you are pricing a decision inside a hand rather than a ticket, the poker pot odds calculator handles the equivalent break-even calculation. And for a pure probability-of-compounding problem with no bookmaker involved — the chance that a rare event happens at least once across many independent trials — the drop rate probability calculator uses the same multiplication rule in reverse.
Parlays remain popular because the multiplication genuinely does produce large payouts on small stakes, and there is nothing dishonest about buying a lottery-shaped payoff knowingly. The point of this calculator is to make the price of that shape explicit, so that the choice is informed rather than assumed. If you want the arithmetic of a genuinely lottery-shaped bet, the lottery odds calculator shows what the extreme version of the same trade looks like.