Calculator

Parlay Calculator

About this calculator

A parlay ties several bets together. Every leg has to win, the odds multiply, and so does the cut.

Parlays are the classic −EV trap. See the full breakdown in Betting Mistakes →

A parlay combines several legs into one bet that pays only if every leg wins. The payouts multiply, but so do the win probabilities, and that second part is what the ticket does not advertise.

Where the money goes

Three legs at −110 combine to about 6.96, so $100 comes back as $695.79. That sounds enormous right up until you notice the break-even chance is 14.37%.

The problem is that the margin multiplies too. One −110 bet gives up about 4.5%. Three of them tied together gives up about 12.7%, on one ticket.

None of which makes a parlay a mistake. It makes it an expensive way to buy variance, and the price of that variance is printed on the ticket if you work it out.

Legs that depend on each other break this arithmetic entirely, in both directions. That one is below.

Want more math?the Nerd Corner

d_{\text{parlay}} = \prod_{i=1}^{n} d_i, \qquad \text{payout} = S \cdot d_{\text{parlay}}, \qquad p_{\text{break-even}} = \dfrac{1}{d_{\text{parlay}}} = \prod_{i=1}^{n} q_i

where:

$n$ : the number of legs on the ticket (2 or more).
$d_i$ : the decimal odds of leg i, total return per dollar including stake. American −110 is $d = 1.909$ ; +180 is $d = 2.80$ .
$q_i$ : the implied probability of leg i, $1/d_i$ , vig included.
$S$ : the stake in dollars.
$d_{\text{parlay}}$ : the combined decimal odds; the payout is $S \cdot d_{\text{parlay}}$ and the profit $S(d_{\text{parlay}} - 1)$ .
$p_{\text{break-even}}$ : the hit rate the ticket needs to lose nothing over the long run.

Worked at the form’s defaults (three −110 legs on a $100 stake):

d_{\text{parlay}} = 1.909^3 = 6.958 \;\Rightarrow\; +596, \qquad \text{payout} = \$100 \times 6.958 = \$695.79

p_{\text{break-even}} = 0.5238^3 = 14.37\% \;\text{ vs a fair } 0.5^3 = 12.5\%

How a parlay is priced

Convert each leg to decimal odds (what $1 returns, stake included), multiply them, then multiply by the stake for the payout. The break-even chance (how often the ticket must hit just to not lose money) is the reciprocal of that product, which is the same thing as multiplying every leg’s implied probability. Both identities are one statement seen from two sides, which is why a parlay quote can never be read on its own: +596 is only informative next to 14.37%.

Why the house edge compounds

Every posted price carries the book’s cut (the vig). A single −110 leg holds 4.55%: the two sides imply 52.38% each, 104.76% in total. Multiply the legs and the margins multiply too. Set the combined price against the fair price the same legs would carry with no vig ( $2^n$ for coin flips) and the hold falls out exactly:

\text{hold} = 1 - \dfrac{d_{\text{parlay}}}{d_{\text{fair}}}, \qquad d_{\text{fair}} = \prod_{i=1}^{n} \dfrac{1}{p_i}

Legs (all −110)	Combined decimal	American	Break-even chance	Hold
1 (a straight bet)	1.909	−110	52.38%	4.55%
2	3.645	+264	27.44%	8.88%
3	6.958	+596	14.37%	13.03%
4	13.283	+1228	7.53%	16.98%
5	25.359	+2436	3.94%	20.75%
6	48.413	+4741	2.07%	24.36%
10	643.08	+64208	0.16%	37.20%

One number to sit with: the hold on a ten-leg ticket of standard −110 legs is 37.2%, worse than most casino games and built out of the fairest prices on a sportsbook’s board. Nothing is hidden in the multiplication: the margin simply rides along on every leg. Priced against three true coin flips, the default ticket returns 0.125 × 6.958 − 1 = −13.03% per dollar, against −4.55% on a single −110 leg.

Common mistakes

Reading the payout without the break-even. +596 sounds generous until it is placed next to 14.37%. Price and probability are the same number written twice.
Assuming the legs are independent. Same-game legs are correlated. Books quote a shorter price than the product; this page shows the product, the benchmark that adjustment is measured against.
Mistaking the break-even chance for the true chance. The break-even figure has the vig baked in. The true chance comes from de-vigging each leg first.
Trusting the advertised combined odds. Some books round the multiplied decimal down before quoting it, so the posted price and the product can differ.

Check this number yourself

moneyline_to_odds and moneyline_to_prob are public functions in the MIT-licensed teachersbettextbook package; the parlay is one multiplication on top of them:

pip install teachersbettextbook

from teachersbettextbook import moneyline_to_odds, moneyline_to_prob
legs = (-110, -110, -110)
d = 1.0
for ml in legs:
    d *= 1 + moneyline_to_odds(ml)   # 6.9579...
payout    = 100 * d                  # 695.79
breakeven = 1 / d                    # 0.14372

Same inputs, same output, no account required. How the open and hosted halves split →

The reverse is also true: independent legs that each carry an edge multiply the edge, not just the margin. Enter your real win chance above and the calculator says whether the ticket clears zero. No edge prints $0.

Size any bet with the Kelly calculator, check a single leg with the +EV calculator, or read the full walkthrough in Betting Mistakes.

Frequently asked questions

Why do parlays carry a bigger house edge?

Because the vig compounds. Each leg's price already carries the book's margin, and multiplying the legs multiplies the margins. Three −110 legs each hold 4.55%, but the three-leg ticket holds 13.03%: the combined price is 6.96 where three fair coin flips would pay 8.00. The house edge is collected once per leg, all on one ticket.

How is a parlay payout calculated?

Multiply the decimal odds of every leg to get the combined decimal odds, then multiply by the stake. A $100 three-leg parlay of −110s pays 1.909 × 1.909 × 1.909 = 6.958 times the stake, or $695.79 back for $100 risked, a profit of $595.79, quoted as +596. The break-even chance is the reciprocal: 14.37%.

Does this calculator work for same-game parlays?

The arithmetic assumes the legs are independent, which same-game legs are not: a quarterback's passing yards and his team's win are correlated. Books price same-game parlays with a correlation adjustment, so the posted price is usually shorter than the product of the legs. Use this page to measure that gap, not to reproduce the book's number.