Every Leg Must Win
How parlays work in sports betting
According to Etymonline [1], the term parlay dates back to 1701, originating in the card game faro. It comes from the French paroli and describes the practice of applying money won to a continuing bet. The concept eventually found its way into sports betting [2], but the basic idea remained the same.
A parlay is simply a bet in which the winnings from one bet are rolled over into the next.
Instead of placing two consecutive bets and reinvesting the winnings from the first into the second, you combine them into a single parlay. Better yet, the games can take place simultaneously. Every leg must win for the parlay to pay out. A small stake can produce a large payout, and the number of legs lets you shape your risk-reward profile. That is probably why people like them.
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Each bet in a parlay is called a leg. If the legs of your parlay are independent, the probability of winning the parlay is the product of the individual probabilities. For example, you would consider the legs to be independent when betting moneyline (who wins?) on different NHL games on the same day. In contrast, if your bet includes moneyline and half-time winner of the same team then the legs are no longer independent and we cannot use this simple formula. This is an example of a “same-game parlay”. Traditional bookmakers often refuse them because correlated legs are hard to price, and the apps that do offer them usually charge a larger margin.
Suppose every leg is a coin toss (50% chance, all legs independent). Decimal odds are simply 1 / probability: 2.00 doubles your money, 4.00 quadruples it. Here’s how quickly the payout grows, and the chance of winning shrinks, as you add legs:
| Number of Legs | Fair Probability | Fair Decimal Odds |
|---------------:|-----------------:|------------------:|
| 1 | 50.0000% | 2.00 |
| 2 | 25.0000% | 4.00 |
| 3 | 12.5000% | 8.00 |
| 4 | 6.2500% | 16.00 |
| 5 | 3.1250% | 32.00 |
| 6 | 1.5625% | 64.00 |These are fair odds assuming no bookmaker markup (no-vig [3]). In practice, a bookmaker won’t offer 2.00 on a coin flip. A typical price is −110 (about 1.91), which means about a 4.5% edge on a single bet. That’s the 4.76% overround from [3], expressed per dollar staked. Parlay odds are calculated by multiplying the leg odds, so the margin multiplies too. A six-leg parlay at −110 pays about 48.4 instead of the fair 64, so the bookmaker’s edge has grown from about 4.5% to about 24%. That is why parlays are among a bookmaker’s most profitable products, and why de-vigging each leg matters. Once you have each leg’s fair probability (that’s what HalfPoint’s consensus prices are for), you can tell whether a parlay’s price is good value or just a big number.
[1] https://www.etymonline.com/word/parlay
[2] https://en.wikipedia.org/wiki/Parlay
[3] https://halfpoint.medium.com/the-curious-case-of-the-missing-4-7-8895fe4667ed
