How Parlay Odds Are Calculated (and Why the Vig Compounds)
Related book: The Sports Investing Blueprint by Ethan Cole · Amazon
A parlay is one bet on two or more prices, and it pays only if every leg wins. Convert each American price to a decimal, multiply those decimals, and convert the product back. The margin is already inside each leg, and multiplication keeps it in the product, so the gap between the posted payout and a fair payout grows as legs are added.
Every price below is hypothetical. These are not odds from a book, not a game on today’s card, and not a suggestion to bet any side. This page is not an automatic calculator. A larger payout is not an edge, and it is not a profit. Percentages, decimals, and dollar amounts are the exact result rounded half up to two decimal places. Do not multiply already rounded numbers and treat that as exact. The implied probability and the vig page rounds a single market the same way. Other hypothetical American prices, with the same rounding, are listed in the American odds to implied probability table. Multiply the exact decimals, not the rounded display, if you stack a price from that table.
From an American price to a decimal
Decimal odds are the total return on a one-unit stake, including the stake, if that one bet wins.
For a negative American price, drop the minus and call that number A. A price of -110 means A is 110.
Decimal = (A + 100) / A
For a positive American price, call the posted number A. A price of +150 means A is 150.
Decimal = (A + 100) / 100
The implied probability on the vig page is the break-even win rate of that price alone. The decimal is 1 divided by that rate. For -110, the implied probability is 110 / 210, which is 11 / 21, and the decimal is 21 / 11.
| Price | Exact decimal | Displayed decimal | Exact implied probability | Displayed |
|---|---|---|---|---|
| -110 | 21 / 11 | 1.91 | 11 / 21 | 52.38% |
| +150 | 250 / 100 | 2.50 | 100 / 250 | 40.00% |
+150 is only the positive formula. The parlays below do not use it. 21 / 11 is 1.9090…, displayed 1.91. 250 / 100 and 40.00% are exact.
Multiply the legs, then convert back
The parlay decimal is the product of the leg decimals. Each -110 decimal is 21 / 11.
Two legs: (21 / 11) × (21 / 11) = 441 / 121.
Three legs: (21 / 11) × (21 / 11) × (21 / 11) = 9261 / 1331.
441 / 121 is 3.6446…, displayed 3.64. 9261 / 1331 is 6.9579…, displayed 6.96.
The rounded one-leg decimal does not multiply back to those figures. 1.91 × 1.91 = 3.6481, which would display as 3.65, not 3.64. 1.91 × 1.91 × 1.91 = 6.9678…, which would display as 6.97, not 6.96. The tables use the exact product, then round once.
When the decimal D is greater than 2, the American price is (D − 1) × 100. That is the profit on a $100 stake. Both parlays here are above 2, so both prices are positive.
| Parlay | Exact decimal | Displayed decimal | Exact profit on $100 | Displayed profit | Displayed American price |
|---|---|---|---|---|---|
| Two -110 legs | 441 / 121 | 3.64 | 32000 / 121 | $264.46 | +264.46 |
| Three -110 legs | 9261 / 1331 | 6.96 | 793000 / 1331 | $595.79 | +595.79 |
+264.46 is the hypothetical product rounded to a cent. Total return includes the stake: $364.46 for two legs and $695.79 for three. The rounded decimal 3.64 times 100 is $364.00, 46 cents under that two-leg return.
The implied probability of the parlay price is 1 divided by the parlay decimal. It is also the product of the leg implied probabilities.
| Parlay | Exact implied probability | Displayed |
|---|---|---|
| Two -110 legs | 121 / 441 | 27.44% |
| Three -110 legs | 1331 / 9261 | 14.37% |
27.44% and 14.37% are break-even rates for the parlay price alone. They are not the chance that the legs all win.
Posted payout against the fair payout
On a -110 / -110 market, the proportional no-vig method leaves 50.00% and 50.00%. The prices match, so each fair share is exactly one half. The fair decimal for one half is exactly 2. That 50% is not a prediction. It is the -110 price with this margin removed.
If the legs are independent, the fair joint probability is the product of the fair shares. Two legs: 1 / 4, displayed 25.00%. Three legs: 1 / 8, displayed 12.50%. The fair parlay decimal is 4 for two legs and 8 for three. The fair profit on $100 is then exact.
| Parlay | Fair decimal | Fair profit on $100 | Posted profit | Gap, fair minus posted |
|---|---|---|---|---|
| Two -110 legs | 4 | $300.00 | $264.46 | $35.54 |
| Three -110 legs | 8 | $700.00 | $595.79 | $104.21 |
The two-leg gap is 4300 / 121, displayed $35.54. The three-leg gap is 138700 / 1331, displayed $104.21. The posted parlay pays less than the fair product of the no-vig legs, and the third leg widens that dollar gap.
The same prices can be read as probabilities. The gap is the posted implied probability minus the fair joint probability. The relative margin is the posted implied probability divided by the fair probability, minus 1.
| Legs | Fair joint | Posted implied | Gap in percentage points | Relative margin |
|---|---|---|---|---|
| One -110 | 50.00% | 52.38% | 2.38 | 4.76% |
| Two -110 | 25.00% | 27.44% | 2.44 | 9.75% |
| Three -110 | 12.50% | 14.37% | 1.87 | 14.98% |
The one-leg row is the single -110 price against its fair half, not a parlay. The relative margin is (11 / 21) / (1 / 2) − 1 = 1 / 21, displayed 4.76%. That is also the two-way overround on a -110 / -110 market, written as a ratio. It is not a second fee on top of the 2.38 point gap.
Two legs: (22 / 21) × (22 / 21) − 1 = 43 / 441, displayed 9.75%. Three legs: 1387 / 9261, displayed 14.98%. Each added leg multiplies the ratio. The percentage-point gap does not show the compounding by itself. It moves from 2.38 to 2.44 and then to 1.87, because the fair joint shrinks faster than the extra margin grows. The payout gap and the relative margin do: 4.76%, then 9.75%, then 14.98%, and $35.54, then $104.21, on $100 in these hypotheticals.
Do not add the 4.76 point overround once per leg and call that the parlay margin. That overround counts both sides of one market. A parlay price is only the combination. This page does not invent prices for the other combinations.
Correlated legs and same-game parlays
Those fair joints treat the legs as independent: one result does not change the fair share of the next. Separate games are often close enough to that picture for the product to be a useful comparison.
Legs in the same game often move together. A side and a total, or a team and a player on that team, can be decided by the same plays. The fair chance that both happen is then not the product of the separate fair shares. It can be higher or lower, depending on the tie.
Books commonly restrict those combinations, or post a price that is not the multiplied decimal. A same-game parlay is several legs from one game sold as one bet. The product can be too high or too low once the legs are tied, so a book may refuse the pair or price it on its own. This page does not describe any book’s rules, limits, or menu. Correlation does not remove the vig inside each leg. It changes the fair joint, and 25.00% or 12.50% may be the wrong benchmark.
Price, stake, and the usual result
A better price on one leg changes the product, because that decimal is one factor. Line shopping compares legal prices before you take one. Shopping a leg does not make the parlay a pick. The vig page shows that -105 breaks even at a lower rate than -110. That would enter as a larger decimal.
The parlay is one stake, not a stake on each leg. The $100 above is the amount at risk on the combination. Bankroll management and unit sizing asks how large that stake is next to a separate bankroll. A profit of $264.46 on $100 does not say whether $100 is a small fraction of the bankroll. The larger payout is not a reason to raise the stake. What shrinks as legs are added, in this independent sketch, is the fair joint rate: 50.00%, then 25.00%, then 12.50%.
Understanding variance is the limit in results. A parlay at these fair shares loses on most trials even before the vig. A short winning run does not show that the posted price was fair. The 9.75% or 14.98% figure is a fact about these prices.
The bankroll and line-shopping worksheet holds single-price hypotheticals: a flat stake, a stake-to-win, and a fixed or resized streak. It does not multiply legs, and it has no formulas. It can still hold the one stake you chose. It will not turn a parlay price into a fair price.
Bottom line
For two hypothetical -110 legs the posted profit on $100 is $264.46 and the fair profit is $300.00. For three legs the posted profit is $595.79 and the fair profit is $700.00. The relative margin goes from 4.76% to 9.75% to 14.98%. Round the exact figure once. If the legs are correlated, that fair product is the wrong benchmark. None of these numbers is a pick, a stake, or a promise of profit.
Educational content. Hypothetical prices only. No pick and no guarantee of profit. A parlay price is not an edge. 18+/21+ where required. Only where legal. Responsible Gambling