Systems and combinations
System bet 3/5: all 10 combinations and the payout
A 3/5 system splits a five-selection ticket into 10 trebles, so a stake of 100 puts 10 on each. It starts paying with 3 winners, but with exactly 3 winners the ticket is only in profit if the product of those three odds is above 10.
Contents
Key terms
- 3/5 system
- A five-selection ticket split into every combination of three selections. How any K/N system is calculated is explained in How system bets work.
- Combination
- A small ticket of three selections. It pays only if all three win.
- Winner
- A selection that won.
All 10 combinations
Odds of 1.80 / 2.00 / 1.70 / 2.20 / 1.90 and a stake of 100, so each combination gets 10.
| Combination | Odds | Payout |
|---|---|---|
| 1.80 · 2.00 · 1.70 | 6.120 | 61.20 |
| 1.80 · 2.00 · 2.20 | 7.920 | 79.20 |
| 1.80 · 2.00 · 1.90 | 6.840 | 68.40 |
| 1.80 · 1.70 · 2.20 | 6.732 | 67.32 |
| 1.80 · 1.70 · 1.90 | 5.814 | 58.14 |
| 1.80 · 2.20 · 1.90 | 7.524 | 75.24 |
| 2.00 · 1.70 · 2.20 | 7.480 | 74.80 |
| 2.00 · 1.70 · 1.90 | 6.460 | 64.60 |
| 2.00 · 2.20 · 1.90 | 8.360 | 83.60 |
| 1.70 · 2.20 · 1.90 | 7.106 | 71.06 |
| Total | 703.56 |
When all five selections win, all 10 combinations pay, 703.56 in total. The weakest combination returns 58.14 and the strongest 83.60: both are below the stake of 100.
How many winners you need
With h winners, C(h, 3) combinations pay: 1 with 3 winners, 4 with 4 and all 10 with 5.
- With exactly 3 winners one combination with a stake of 10 pays. The ticket is in profit only if the product of those three odds is above 10; with equal odds each must be above 2.154.
- With 4 winners four combinations pay. With equal odds the ticket is in profit when each price is above 1.357.
All five selections have the same odds, so the threshold is easy to see. The stake is 100.
| Odds | 3 winners | Net | 4 winners | Net |
|---|---|---|---|---|
| 1.30 | 21.97 | −78.03 | 87.88 | −12.12 |
| 1.50 | 33.75 | −66.25 | 135.00 | +35.00 |
| 1.80 | 58.32 | −41.68 | 233.28 | +133.28 |
| 2.00 | 80.00 | −20.00 | 320.00 | +220.00 |
| 2.10 | 92.61 | −7.39 | 370.44 | +270.44 |
| 2.20 | 106.48 | +6.48 | 425.92 | +325.92 |
With typical odds between 1.50 and 2.10, a 3/5 system with three winners returns part of the stake but is still at a loss. A profit needs four winners.
Chance of winners
Every selection wins with the same probability p, independently of the others, so the number of winners follows a binomial distribution.
| p | Exactly 3 | Exactly 4 | All 5 | 3 or more |
|---|---|---|---|---|
| 0.45 | 27.57% | 11.28% | 1.85% | 40.69% |
| 0.50 | 31.25% | 15.63% | 3.13% | 50.00% |
| 0.55 | 33.69% | 20.59% | 5.03% | 59.31% |
With p = 0.50 and odds of 2.00 the ticket pays something 50.00% of the time, but it is in profit only with 4 or 5 winners, which is 18.75% of the time.
How much comes back on average, and why a system loses less than an accumulator of the same selections, is shown in Accumulator or system bet.
The system bet calculator with this example works out the payout for every number of winners and for a void selection.
Limitations
- The binomial distribution assumes the same probability for every selection and independent matches. Two bets on the same match are not independent.
- Bookmakers often set a minimum stake per combination and a maximum payout per ticket.
- Rules for postponed and abandoned matches differ between bookmakers.
- The example ignores tax on winnings.