# What does my round robin pay?

Computes the number of parlays in a round robin bet of 3 to 10 picks, the total stake, the payout if every pick wins, and the payout for picks that won, pushed or lost.

- Page: https://www.acalculator.org/game/round-robin-calculator
- JSON spec: https://www.acalculator.org/game/round-robin-calculator.json
- Version: 64b76cb32126

## Default answer

Example with the default inputs (Odds format American, Picks [Odds -110, Result Open or won; Odds -110, Result Open or won; Odds -110, Result Open or won], Picks per parlay 2, Stake per parlay $10.00): A round robin of 3 parlays at $10.00 each costs $30.00 and pays $109.34 if every pick wins.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| format | Odds format | The format of every pick’s odds: American, decimal, or fractional. |
| picks | Picks | One row per pick: its odds and its result so far. |
| size | Picks per parlay | How many picks each parlay holds: 2 for “by 2s”, 3 for “by 3s”, and so on. |
| stake | Stake per parlay | The amount bet on each parlay of the round robin. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| bets | Parlays | The number of parlays: C(picks, size) = picks! ÷ (size! × (picks − size)!). |
| totalStake | Total stake | Stake per parlay × number of parlays. |
| maxPayout | Payout if every pick wins | Stake per parlay × the sum of every parlay’s decimal odds. |
| maxProfit | Profit if every pick wins | Payout if every pick wins minus the total stake. |
| payout | Payout with these results | The payout counting lost parlays as 0 and dropping pushed picks from the parlays they are in. |
| profit | Profit with these results | Payout with these results minus the total stake. |

## Method

Parlays = C(n, k) for n picks by k; each parlay pays stake × the product of its decimal odds (pushed picks drop out, a lost pick makes it pay 0); the round robin pays the sum.

## Assumptions

- Each parlay is a separate bet with the same stake, and its odds are the product of its picks’ decimal odds.
- A pushed pick drops out of every parlay it is in; a parlay whose picks all pushed returns its stake.
- Results are mathematics, not betting advice.

## Worked examples

1. format = american, picks = {"odds":"-110","result":"open"} or {"odds":"-110","result":"open"}, size = 2, stake = $10.00 gives bets = 3, totalStake = $30.00, maxPayout = $109.34, maxProfit = $79.34. Source: hand calculation in content.mdx: 3 × 10 × 441/121 = 13230/121.
2. format = american, picks = {"odds":"-110","result":"open"} or {"odds":"-110","result":"open"}, size = 2, stake = $10.00 gives payout = $36.45, profit = $6.45. Source: hand calculation in content.mdx: only the parlay of the two winners pays, 10 × 441/121.
3. format = decimal, picks = {"odds":"2","result":"open"} or {"odds":"3","result":"open"}, size = 3, stake = $5.00 gives bets = 4, totalStake = $20.00, maxPayout = $315.00, maxProfit = $295.00, payout = $112.50, profit = $92.50. Source: hand calculation in content.mdx: all win 5 × (24 + 9 + 12 + 18) = 315; with the push 5 × (6 + 9 + 3 + 4.5) = 112.5.

## FAQ

### How many parlays are in a round robin?

The number of ways to choose the parlay size k from n picks: C(n, k) = n! ÷ (k! × (n − k)!). Three picks by 2s make 3 parlays, four picks by 2s make 6, and five picks by 3s make 10.

### How much does a round robin cost?

The stake per parlay times the number of parlays. Four picks by 2s at 10 each is 6 parlays, so 60 in total.

### How is a round robin paid?

Each parlay is settled on its own: it pays stake × the product of its picks’ decimal odds, or nothing if one of its picks lost. The round robin pays the sum. So it can still pay when some picks lose, unlike a single parlay of every pick.

### What happens when a pick pushes?

The pushed pick drops out of each parlay it is in, and those parlays pay on their other picks. A 2-pick parlay with one push pays like a single bet on the other pick.

### Is a round robin better than a parlay?

It is a different bet, not a better one. It spreads the stake over many smaller parlays: it can pay when a pick loses, but it costs more in total and pays less than one parlay when every pick wins. Each parlay still carries the bookmaker’s margin.

## Sources

- Wikipedia, Parlay (gambling), round robin: https://en.wikipedia.org/wiki/Parlay_(gambling)
- Grinstead and Snell, Introduction to Probability, 2nd ed., American Mathematical Society, section 3.2, combinations (binomial coefficients): https://math.dartmouth.edu/~prob/prob/prob.pdf
- National Council on Problem Gambling, help and treatment (1-800-MY-RESET): https://www.ncpgambling.org/help-treatment/