Parlay Math at minus 110: How the Vig Compounds From 4.55% on One Leg to 37.20% on Ten
A single bet at minus 110 on a coin flip line costs 4.55% of the stake in expected value. Put two of those bets into a parlay and the cost is 8.88%. Four legs, 16.98%. Ten legs, 37.20%. The vig does not stay at one leg's worth when bets are combined. It compounds, and the reason is the same multiplication that makes the payout look attractive.
The concept
A parlay pays only if every leg wins. In exchange, its price is the product of the individual prices. At true odds pricing, which many US sportsbooks use for standard parlays, a two leg parlay at minus 110 and minus 110 pays the decimal price of one leg multiplied by the other.
The vig is built into each of those decimal prices. Multiplying the prices multiplies the vig along with them.
The formula
Convert each leg to decimal first. For a negative American price:
decimal = 1 + 100 / |American price|
At minus 110, that is 1 + 100/110 = 1.9091.
For a parlay of n legs at the same price, the parlay's decimal price is 1.9091 raised to the power n. The expected value per $1 staked, when each leg is a genuine 50% proposition, is:
EV = (0.5 x 1.9091)^n − 1 = 0.95455^n − 1
The figure 0.95455 is the share of value kept on one leg after the vig. Raising it to the power n is where the compounding happens.
Worked example: two legs
Two NFL sides, each minus 110, each assumed to be a true 50% chance.
- Parlay price. 1.9091 x 1.9091 = 3.6446 decimal.
- American conversion. For a decimal price above 2.00, American = (decimal − 1) x 100, so (3.6446 − 1) x 100 = +264.
- Implied probability. 1 / 3.6446 = 27.44%.
- True probability. 0.5 x 0.5 = 25.00%.
- Fair price. A 25% event at fair odds pays 1 / 0.25 = 4.00 decimal, or +300.
- Expected value on $100. Win 25% of the time for a profit of $264.46, lose 75% of the time for −$100. That is 0.25 x 264.46 − 0.75 x 100 = 66.12 − 75.00 = −$8.88.
The vig on the parlay, −$8.88 per $100, is almost exactly double the −$4.55 on a single minus 110 bet. Not quite double, because 0.95455 squared is 0.91116, a loss of 8.88% rather than 9.09%.
The compounding table
Same assumptions: every leg is minus 110 and a true coin flip. Fractional prices are rounded to two decimals.
| Legs | Decimal | American | Fractional | Fair price | Expected loss per $100 |
|---|---|---|---|---|---|
| 1 | 1.9091 | minus 110 | 10/11 | +100 | $4.55 |
| 2 | 3.6446 | +264 | 2.64/1 | +300 | $8.88 |
| 3 | 6.9579 | +596 | 5.96/1 | +700 | $13.03 |
| 4 | 13.2833 | +1228 | 12.28/1 | +1500 | $16.98 |
| 6 | 48.4127 | +4741 | 47.41/1 | +6300 | $24.36 |
| 10 | 643.0816 | +64208 | 642.08/1 | +102300 | $37.20 |
Two things stand out. First, the expected loss grows with every leg, by about 4 points per leg at first and more slowly later, because each extra leg takes 4.55% of what is left rather than 4.55% of the original stake. Second, the gap between the offered price and the fair price widens fast. At ten legs the book pays +64208 on an event whose fair price is +102300.
The common error
The usual mistake is to judge a parlay by its payout rather than its price. +264 on two legs looks generous next to minus 110 on one. The question that matters is what the true probability of the combined outcome is, and how the offered price compares with the fair price for that probability. On a two leg minus 110 parlay, the book implies 27.44% for an outcome that is 25.00% if the legs are coin flips. That 2.44 point gap is the vig, expressed as probability.
A second error is to assume legs are independent when they are not. Same game parlays combine outcomes that move together, such as a team's side and the game total. Sportsbooks price that correlation into the parlay, so the simple multiplication above does not apply to them, and their effective margin cannot be read from the leg prices.
State note
Sports betting is legal and regulated state by state, and each state's regulator sets what operators may offer. Parlay pricing itself is set by the operator, and some books use fixed payout cards instead of true odds pricing. A fixed card paying less than the multiplied price raises the expected loss above the figures in the table. Compare the offered parlay price with the product of the leg prices before assuming true odds.
What the math does not tell you
The table assumes every leg is a true 50% proposition. Real lines are not coin flips, and the true probability of any leg is unknown. What the math does show is fixed regardless: every leg carries its own vig, and combining legs multiplies them. No selection of legs changes that structure.
Responsible gambling
Legal sports betting age is 21 in most states. Parlays concentrate a small stake into a long price, which makes a losing run easy to underestimate. Set a deposit limit with your operator before betting, and if gambling stops feeling manageable, the National Council on Problem Gambling's helpline is free and confidential on 1 800 522 4700.