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Round Robin Calculator

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By Alex Thornton · author of the MIT-licensed teachersbettextbook library · Last reviewed 24 July 2026

A round robin turns one list of picks into a pile of separate parlays. Here is how many, and what they really cost.

Same math on a single ticket: the parlay calculator →

A round robin turns a list of picks into every parlay of a chosen size, and each parlay is its own bet with its own stake. The count grows fast, and so does the total at risk.

The number the slip does not show

Three legs combined by twos is three two-leg parlays, each staked and settled on its own. Not one bet. Three.

So the stake is per ticket, and the real risk is that stake times the ticket count. A “$10 round robin” on four legs by twos risks $60, not $10.

And the margin is paid once per ticket. Six tickets means six helpings of vig on the same handful of opinions.

Legs that move together break the arithmetic completely, because the tickets stop being independent. Detail below.

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T = \binom{n}{k} = \dfrac{n!}{k!\,(n-k)!}, \qquad R = S \cdot T, \qquad \text{cashing at } j \text{ correct} = \binom{j}{k}

where:

$n$ : the number of selections, 3 or more. At $n = k$ it is an ordinary parlay.
$k$ : the combination size, 2 for “by twos”, 3 for “by threes”. By twos and threes is $\binom{n}{2} + \binom{n}{3}$ parlays.
$T$ : the ticket count, dimensionless.
$S$ : the stake per ticket, in dollars (the field people misread). The slip charges $S$ for every combination, not $S$ in total.
$R$ : the total risk, in dollars.
$j$ : how many of the $n$ legs won. A ticket cashes only if all $k$ of its legs are among those $j$ , so $\binom{j}{k}$ tickets cash.
$d_i$ : the decimal odds of leg $i$ , total return per dollar including stake. −110 is $d = 1.9091$ .

The return when exactly $j$ legs win, when every leg carries the same price:

\text{return}(j) = \binom{j}{k} \cdot S \cdot d^{\,k}, \qquad \text{net}(j) = \binom{j}{k} \cdot S \cdot d^{\,k} - S\binom{n}{k}

Worked example: the numbers in the form below

Three legs at −110, combined by twos, $10 a ticket. Each −110 is decimal 1.909091.

T = \binom{3}{2} = 3 \;\Rightarrow\; R = 3 \times \$10 = \$30, \qquad d^{2} = 1.909091^{2} = 3.644628

\text{each winning ticket pays } \$10 \times 3.644628 = \$36.45

j = 3:\ 3 \text{ tickets} \Rightarrow \$109.34 \;(\!+\$79.34), \quad j = 2:\ 1 \Rightarrow \$36.45\;(\!+\$6.45), \quad j \le 1:\ 0 \Rightarrow -\$30

Two legs right on three selections shows +$6.45, against +$8.18 for the same $30 on three straight bets. The round robin is behind there and ahead only when all three land: +$79.34 against +$27.27.

How a round robin is priced

Nothing new happens at the pricing layer. Each combination is an ordinary parlay:

d_T = \prod_{i \in T} d_i, \qquad \text{payout}(T) = S \cdot d_T

The new object is the hit-level structure: a straight bet and a parlay have two outcomes each, a round robin has one per hit level, and the cashing-ticket count is a binomial coefficient of a binomial coefficient:

\text{cashing}(j) = \binom{j}{k}, \qquad \binom{j}{k} = 0 \text{ whenever } j < k

The payoff curve is therefore convex: on a 6-selection card by twos, going from 3 correct to 4 takes the cashing tickets from 3 to 6.

The vig is paid once per ticket

Expected value is additive: the round robin’s is the sum of its tickets’, and every ticket is a $k$ -leg parlay:

\mathbb{E}[\text{round robin}] = \sum_{T} S\left(\textstyle\prod_{i \in T} p_i \cdot d_T - 1\right)

With one price and one true probability across the legs this collapses: the round robin’s expected value per dollar equals the $k$ -leg parlay’s, exactly, for any number of selections. Three −110 legs on coin flips:

\mathbb{E} = 3 \times \$10 \left(0.5^{2} \times 3.644628 - 1\right) = -\$2.6653 \;\Rightarrow\; -8.88\%\ \text{per \$1}

Feed it eight selections instead of three and the tickets go from 3 to 28 and the risk from $30 to $280, while the expected value per dollar stays at −8.88%, against 4.55% for three straight bets and 13.03% for one three-leg parlay on the same money. The round robin sits between them because its tickets are two-leg parlays: the ticket length did everything, the structure did nothing.

How many legs have to land

The break-even hit level is the smallest $j$ whose cashing tickets outrun the whole card:

j^{*} = \min\left\{ j : \binom{j}{k} \cdot d^{\,k} > \binom{n}{k} \right\}

The stake cancels, so $j^{*}$ depends only on the count of selections and the price, not on the stake.

$j^{*}$ climbs with the size of the card (2 of 3, 4 of 6, 5 of 8) because risk grows faster than the cashing count, and it is a step function: clearing the bar by one leg pays +$8.68 on seven selections and +$84.46 on eight.

What a round robin cannot do

It cannot manufacture an edge the legs do not have: expected value is additive, so if the legs are negative-EV every ticket is too. Finding an edge is a separate calculation: expected value per leg, de-vigged, then Kelly. Enter your own probability and the panel prints $0 when the inputs contain no edge.

Correlated legs break the arithmetic

Everything above multiplies probabilities, which assumes independent legs. Same-game legs are not: a team’s moneyline and its quarterback’s passing yards move together. Positive correlation fattens the all-hit and all-miss outcomes and thins the middle hit levels, exactly the region a round robin was meant to cover. Books apply a correlation adjustment, so this page’s product will not match a posted same-game price.

Common mistakes

Reading the stake field as the total. $20 a ticket on six selections by twos is $300 at risk; the multiplier is $\binom{n}{k}$ and it grows fast.
Treating a high hit rate as an edge. The default card shows a profit half the time and still loses 8.88¢ per dollar; frequency of profit and expected profit are unrelated.
Assuming partial credit starts at one leg. Below $k$ correct legs nothing cashes at all: a round robin by threes returns zero on two of six.
Averaging mixed-price legs. With unequal prices the return at a hit level is a range, not a number: four legs at −120, −105, +140 and +180 by twos at $25 pay anywhere from $89.48 to $168.00 on two of four. Which legs won matters as much as how many.

Check this number yourself

Combinatorics on top of a parlay, and the parlay is one multiplication on top of moneyline_to_odds in the MIT-licensed teachersbettextbook package:

pip install teachersbettextbook

import math
from itertools import combinations
from teachersbettextbook import moneyline_to_odds

legs, k, stake = [-110, -110, -110], 2, 10
tickets = list(combinations(legs, k))
risk    = stake * len(tickets)                       # 30.0
payout  = [stake * math.prod(1 + moneyline_to_odds(m) for m in t)
           for t in tickets]                         # 36.4463 each
ev      = sum(p * 0.5 ** k - stake for p in payout)  # -2.6653

Same inputs, same digits, no account required. What is open and what is not.

Frequently asked questions

What is a round robin bet?

A round robin is a set of parlays generated from one list of selections: n legs and a combination size k produce C(n,k) separate parlays, each staked and settled on its own.

How many parlays are in a round robin?

C(n,k) = n! / (k!(n−k)!). Three legs by twos is 3 parlays; four by twos is 6; six by twos is 15; eight by twos is 28.

How do you calculate a round robin payout?

Multiply the decimal odds inside each combination, multiply by the per-ticket stake, and add the tickets whose legs all won. Three −110 legs by twos at $10 a ticket: each winner pays $36.45, so three of three returns $109.34 on $30 at risk.

Does a round robin reduce the vig?

No. Expected value is additive across tickets, so a round robin’s expected value per dollar equals its k-leg parlay’s at any number of selections: 8.88¢ per dollar for two-leg tickets on −110 coin flips.

How many legs must hit for a round robin to show a profit?

The smallest j where C(j,k) times the ticket payout exceeds the total risk. For −110 legs by twos: 2 of 3, 3 of 4, 4 of 6, 5 of 8. The bar rises with the size of the card.

Is a round robin better than a single parlay?

They are different bets, not better and worse. The single parlay pays more when everything lands and nothing otherwise; the round robin pays less at the top and something at two of three. The mean is set by ticket length.