Systems and combinations
Accumulator or system bet: what you gain and what you give up
An accumulator pays only if every selection wins, but then it pays the most. A system splits the stake into smaller combinations, so it can return money with a loser, but its top payout is lower. Because the margin multiplies with the number of legs in a combination, a system with shorter combinations loses a smaller share of the stake on average.
Contents
Key terms
- Accumulator
- An ordinary ticket on which every selection has to win. The odds multiply.
- K/N system
- A ticket of N selections split into every combination of K selections (see How system bets work).
- Payout per leg
- The share of the stake a selection returns on average at its fair probability, for example 95% when the market margin is about 5%.
Same selections, three tickets
Five selections at 1.80 / 2.00 / 1.70 / 2.20 / 1.90 and a stake of 100 on each ticket: an accumulator (5/5), a 4/5 system (5 combinations of 20) and a 3/5 system (10 combinations of 10).
| What won | Accumulator | 4/5 system | 3/5 system |
|---|---|---|---|
| All 5 | 2,558.16 | 1,342.86 | 703.56 |
| All but 2.20 | 0.00 | 232.56 | 252.34 |
| All but 2.20 and 1.90 | 0.00 | 0.00 | 61.20 |
The accumulator has the biggest payout, but any loser wipes out the whole ticket. The 3/5 system returns part of the stake even with two losers, but when all five win it pays 3.6 times less than the accumulator.
Chance and average return
Every combination of K legs is a small accumulator, so the payouts of its legs multiply. If each leg pays back 95%, a three-leg combination returns 0.95³ = 85.74% and a five-leg one 0.95⁵ = 77.38%. Since all combinations of a K/N system have the same length, the whole system returns on average the same as one combination.
K/N system payout = (payout per leg)^KThe same selections. Assumption: each selection's market pays back 95%, so its fair probability is 0.95 / odds (for example 0.95 / 1.80 = 0.5278), and the selections are independent.
| Ticket | Pays something | In profit | Average payout | Average result |
|---|---|---|---|---|
| Accumulator 5/5 | 3.02% | 3.02% | 77.38% | −22.62% |
| 4/5 system | 18.47% | 18.47% | 81.45% | −18.55% |
| 3/5 system | 49.75% | 18.47% | 85.74% | −14.26% |
The 3/5 system pays something almost every other time, but it is in profit only with 4 or 5 winners. In this example three winners always leave it at a loss, so its chance of profit equals that of the 4/5 system.
What really changes
- Accumulator: the biggest payout, the lowest chance of a return and the largest effect of the margin, because it multiplies across every leg.
- A system with shorter combinations: a return even with a loser, a lower top payout and less margin per combination.
- No ticket type changes the chance that a selection wins. When the selections have negative expected value, so does every ticket built from them; a system only changes by how much.
The system bet calculator with these selections shows the payout for every number of winners; change K to 4 or 5 to compare.
Limitations
- A 95% payout per leg and a proportional fair probability are assumptions of the example. Real margins and how they are spread differ by market.
- The maths assumes independent matches. Two bets on the same match are not independent, which is why bookmakers often do not allow them on one ticket.
- Bookmakers may set a minimum stake per combination and a maximum payout per ticket.
- The example ignores tax on winnings, which can hit an accumulator and a system differently.