Round Robin Calculator

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

A round robin turns one list of selections into a pile of parlays. This page enumerates every combination, totals the risk, and prints the return at each hit level — plus the number the ticket slip never shows: the parlay margin, paid once per ticket.

A round robin is a set of parlays built automatically from a list of selections: n legs and a combination size k produce C(n, k) separate parlays, each staked and settled on its own. Three legs combined by twos is three two-leg parlays, not one three-leg parlay.

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 you hand the book, 3 or more. With $n = k$ there is only one combination, which is an ordinary parlay.
$k$ — the combination size: 2 for “by twos”, 3 for “by threes”. A ticket combined by twos and threes is $\binom{n}{2} + \binom{n}{3}$ parlays.
$T$ — the ticket count, dimensionless.
$S$ — the stake per ticket, in dollars. This is 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 actually won. A ticket cashes only when all $k$ of its legs are among those $j$ , so $\binom{j}{k}$ tickets cash and the rest are dead.
$d_i$ — the decimal odds of leg $i$ , total return per dollar including stake. American −110 is $d = 1.9091$ .

The return when exactly $j$ legs win, for a card where 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 spread over three straight bets. The round robin is behind at that hit level, and ahead only when all three land: +$79.34 against +$27.27. What the structure moves is the dispersion, not the mean — at a true 50% per leg the set returns −8.88% per dollar against the straights’ −4.55%.

Same math on a single ticket — the parlay calculator →

How a round robin is priced

Nothing new happens at the pricing layer. Each combination is an ordinary parlay, priced the way any parlay is priced: multiply the decimal odds of the legs inside it.

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

The only genuinely new object is the hit-level structure. A straight bet has two outcomes and a parlay has two outcomes; a round robin has $n+1$ of them, indexed by how many legs came in. The count of cashing tickets is a binomial coefficient of a binomial coefficient:

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

So the payoff curve is convex in $j$ . Getting one more leg right does not add a fixed amount — going from 3 correct to 4 correct on a 6-selection card by twos takes the cashing tickets from 3 to 6, doubling the return.

The vig is paid once per ticket

This is the number the slip does not print, and the reason a round robin is not the hedge it looks like.

Expected value is additive. The round robin’s expected value 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)

When every leg carries the same price and the same true probability, that collapses to something blunt: the round robin’s expected value per dollar equals the $k$ -leg parlay’s, exactly, for any number of selections. Three −110 legs on genuine 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 the same calculator eight selections instead of three and the ticket count goes from 3 to 28, the risk from $30 to $280, and the expected value per dollar stays at −8.88%. Spreading money across combinations changes the shape of the outcome distribution. It does not change the mean.

$30 on three −110 legs, three ways	3 of 3	2 of 3	1 of 3	0 of 3	EV / $1
Three straight bets, $10 each	+$27.27	+$8.18	−$10.91	−$30.00	−4.55%
Round robin by 2s, $10 × 3 tickets	+$79.34	+$6.45	−$30.00	−$30.00	−8.88%
One three-leg parlay, $30	+$178.74	−$30.00	−$30.00	−$30.00	−13.03%

Read the rows against each other rather than down a single column. The round robin sits between the other two on every measure at once — top payout, floor, and expected loss rate. It is the middle row because its tickets are two-leg parlays: shorter than the three-leg ticket, longer than the singles. The structure did nothing; the ticket length did everything.

Where the −8.88% comes from, outcome by outcome

Three coin-flip legs at −110, by twos, $10 a ticket. At $p = 0.5$ the hit levels are binomial:

P(3) = \tfrac{1}{8},\quad P(2) = \tfrac{3}{8},\quad P(1) = \tfrac{3}{8},\quad P(0) = \tfrac{1}{8}

\mathbb{E} = \tfrac{1}{8}(79.3388) + \tfrac{3}{8}(6.4463) + \tfrac{3}{8}(-30) + \tfrac{1}{8}(-30) = -\$2.6653

The two routes agree to the cent: −$2.6653 by enumerating outcomes, −$2.6653 by summing three identical two-leg parlay EVs. Note the other half of the picture — this card shows a profit half the time (the 1/8 + 3/8 of trials where two or three legs land) while losing 8.88¢ on the dollar. A high hit rate and a negative mean are not in tension; they are the same number seen from different ends.

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 per ticket cancels, so $j^{*}$ depends only on the count of selections and the price. For −110 legs combined by twos:

Selections	Tickets	Risk @ $10	Legs to clear it	Net there	Net if all land
3	3	$30	2 of 3	+$6.45	+$79.34
4	6	$60	3 of 4	+$49.34	+$158.68
5	10	$100	3 of 5	+$9.34	+$264.46
6	15	$150	4 of 6	+$68.68	+$396.69
7	21	$210	4 of 7	+$8.68	+$555.37
8	28	$280	5 of 8	+$84.46	+$740.50

Two things fall out of that column. First, $j^{*}$ climbs with the size of the card — more selections means a higher bar, not a lower one, because the risk grows faster than the cashing count. Second, the “Net there” column is jagged: clearing the bar by exactly one leg pays $8.68 on a seven-selection card and $84.46 on an eight-selection card. The break-even level is a step function, and a card sitting just under a step is paying full price for a payoff it barely reaches.

Second example: four legs, mixed prices, by twos

Legs at −120, −105, +140 and +180, combined by twos at $25 a ticket. Six tickets, $150 at risk.

d = (1.8333,\ 1.9524,\ 2.40,\ 2.80) \;\Rightarrow\; d_T \in [3.5794,\ 6.72]

\text{payout per ticket} = \$25 \times d_T \in [\$89.48,\ \$168.00]

With mixed prices the hit level no longer pins the return: two of four correct pays anywhere from $89.48 (the two shortest legs) to $168.00 (the two longest), against $150 risked. The panel above prints that band rather than a single figure, because a single figure would be a fiction. Which two legs land matters as much as how many.

What the round robin actually changes

Three statements, all arithmetic on the table above, none of them a recommendation:

The mean is fixed by the ticket length. Expected value per dollar equals the $k$ -leg parlay’s, whatever $n$ is. Combining by twos costs 8.88¢ per dollar on −110 coin flips; by threes it costs 13.03¢.
The variance falls. A single parlay pays once or not at all. A round robin has $n+1$ outcomes, and the middle ones are non-zero. That is a genuine change — it is just a change in dispersion, priced at nothing and worth nothing on average.
The total risk multiplies quietly. $10 a ticket on eight selections by twos is $280, not $10. Round-robin slips are the single most common source of “I did not realize I had that much down”, because the stake field reads per ticket and the ticket count is a binomial coefficient.

If the legs carry a real edge, the same additivity works in your favor: every ticket is a positive-EV parlay and the sum is positive too. Finding that edge is a different calculation — expected value on each leg first, de-vigged against a reference price, then Kelly for the size. A round robin cannot manufacture an edge that the legs do not have.

Correlated legs break the arithmetic

Everything above multiplies probabilities, which assumes the legs are independent. Same-game legs are not: a team’s moneyline and its quarterback’s passing yards move together. Correlation cuts both ways in a round robin and it is not symmetric — positively correlated legs make the all-hit and all-miss outcomes fatter and the middle hit levels thinner, which is exactly the region the round robin was supposed to cover. Books price same-game combinations with a correlation adjustment, so the posted price will not match the product of the legs. This page computes the product; the gap between it and the book’s quote is the adjustment, and it is worth measuring rather than assuming.

When to use this calculator

Before the slip is confirmed. The ticket count and the total risk are the two figures most often misread, and both are printed above the fold here.
To compare structures on one bankroll. Enter the same legs and read the three-row table: straights, round robin, single parlay, on identical money.
To find the break-even hit level. “How many do I need?” has an exact answer, and it is usually one leg higher than people guess.
To check a book’s quoted round-robin price against the product of the legs. Any gap is the book’s extra margin or a correlation adjustment.
With an honest probability in the last field. The panel prints expected value on your own estimate, and prints $0 when the inputs contain no edge.

Common mistakes

Reading the stake field as the total. $20 a ticket on a six-selection round robin by twos is $300 at risk. The multiplier is $\binom{n}{k}$ and it grows fast.
Believing the structure dilutes the vig. It does not. Expected value per dollar is the $k$ -leg parlay’s figure, unchanged at any number of selections.
Treating a high hit rate as an edge. The default card above shows a profit half the time and still loses 8.88¢ per dollar. Frequency of profit and expected profit are unrelated quantities.
Assuming partial credit starts at one leg. Below $k$ correct legs, nothing cashes at all. A round robin by threes returns exactly zero on two of six.
Averaging mixed-price legs. With unequal prices the return at a given hit level is a range, not a number — which legs won changes the payout, sometimes by a factor of two.
Multiplying correlated legs. Same-game selections violate the independence the probability arithmetic rests on, and the error lands hardest on the middle hit levels.

Check this number yourself

A round robin is combinatorics on top of a parlay, and the parlay is one multiplication on top of moneyline_to_odds — a public function in the MIT-licensed teachersbettextbook package. The whole card in eleven lines:

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. More on what is open and what is not: open source.

A round robin is a packaging decision, not a pricing one. The price lives in the legs and in how many of them ride on each ticket — which is why the parlay calculator is the page underneath this one, the no-vig calculator is where a leg’s fair price comes from, and expected value is the only place an edge can enter. For the structures that sit next to this one on a bet slip, see the teaser calculator and the middle calculator.

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. Three legs combined by twos is three two-leg parlays, not one three-leg parlay.

How many parlays are in a round robin?

C(n,k) = n! / (k!(n−k)!). Three legs by twos is 3 parlays; four legs by twos is 6; five by twos is 10; six by twos is 15; eight by twos is 28. Combining by twos and threes adds the two counts together.

How do you calculate a round robin payout?

Multiply the decimal odds inside each combination, multiply by the stake per ticket, and add up the tickets whose legs all won. If exactly j of the n legs win, C(j,k) tickets cash. Three −110 legs by twos at $10 a ticket risk $30 and return $109.34 when all three land, $36.45 when two land, and nothing below that.

Does a round robin reduce the vig?

No. Expected value is additive across tickets, so a round robin’s expected value per dollar is the same as the k-leg parlay it is built from. Three −110 legs by twos, at a true 50% per leg, return −8.88% per dollar — identical to a single two-leg parlay, and identical again at five, six or eight selections. What changes is the spread of outcomes, not the mean.

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

The smallest j where C(j,k) × ticket payout exceeds the total risk. For −110 legs combined by twos: 2 of 3, 3 of 4, 3 of 5, 4 of 6, 4 of 7 and 5 of 8. Adding selections raises the number of legs that have to land before the set returns more than it risked.

Is a round robin better than a single parlay?

They are different bets, not better and worse. A single three-leg parlay of −110s returns $178.74 on $30 when all three land and −$30 otherwise. The same $30 as a 3-by-2s round robin returns $79.34 on three, $6.45 on two and −$30 below that. The round robin trades the top payout for a result on partial hits; the expected loss rate moves from −13.03% to −8.88% per dollar because the tickets are shorter, not because the structure is cheaper.