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Summary
I've always had an interest in numbers, although I've never been a gambler. I thought it would be interesting to find optimal row sets. Naturally, the computational time required is enormous. After exploring different search methods and code improvements, sets of rows were found that improve the chances of winning a minimum of 6th division and increase your expected winnings over time. At the bottom of this page you will find a ticket generator that will create number sets for up to ten tickets, which will improve your chances of winning by up to an average of 11.3% over randomly generated rows. It's worth noting that 8 improved tickets have greater coverage than the average of 9 randomly generated ones (see table below).
Improving The Odds (Basic Theory)
Each row on your lotto ticket covers a massive 93,160 (34C1 + 6C5 x 34C2 + 6C4 x 34C3) possible draws. This is because you only need 4 out of the 7 numbers drawn in order to win. There are a total of 18,643,560 (40C7) possible draws. So on the face of it you should only need about 200 rows (18643560/93160) in order to guarantee at least a win in 6th division. The problem is that rows do overlap in their coverage. Take the rows 1 2 3 4 5 6 and 6 7 8 9 10 11 as an example. Winning number sets of both rows include 1 2 3 6 7 8 9, 1 2 3 6 7 8 10, 1 2 3 6 7 8 11, 1 2 3 6 7 9 10, etc. It works out that just two rows with one identical number share 100 possible draws. It gets bad, very bad quickly:
| Identical Numbers | 1 | 2 | 3 | 4 | 5 | 6 |
|---|
| Coverage Overlap | 100 | 1160 | 5616 | 15784 | 54560 | 93160 |
|---|
It's even goes from bad to worse still, because most people buy more than 2 rows and all the rows interact with each other depending on the numbers shared. Beyond 6 rows it is impossible to avoid coverage overlap as there are only 40 numbers to choose from. The more rows the greater the overlap. By searching for row sets that share less common coverage, you can increase your overall chances of winning. This does however lessen your chances of winning twice on the same set of rows, but it is better to win once on 10 different draws than 10 times on one draw as winnings are divided by the number of matching rows.
| Rows | Random % | Improved % | Gain % |
|---|
| 10 | 4.9 | 5.0 | 2.0 |
| 20 | 9.5 | 9.9 | 4.1 |
| 30 | 14.0 | 14.8 | 6.2 |
| 40 | 18.2 | 19.6 | 7.8 |
| 50 | 22.2 | 24.2 | 9.0 |
| 60 | 26.0 | 28.5 | 9.7 |
| 70 | 29.6 | 32.6 | 10.1 |
| 80 | 33.0 | 36.5 | 10.4 |
| 90 | 36.3 | 40.2 | 10.9 |
| 100 | 39.4 | 43.9 | 11.3 |
| 110 | 42.4 | 47.3 | 11.6 |
Where To From Here
- True optimality
- These sets used a greedy search method where each row was found by looking for the next best row with the least overlap. The problem is that choosing a better row at the start, may prevent you from choosing overall better rows later (although, thus far I have found little evidence to support this). Finding true optimality may require a brute force approach (testing every possible outcome and choosing the best solution) for each set length. This is massive. To brute force 10 rows, 498 octilliard (4.98x1053 (40C6Crows-1)) possibilities need to be checked. Even if I could test one sextilliard (1x1039) possibilities per second (although I average less than 50) it would take nearly 16 million years. So the trick is to find ways of reducing the search set and still guarantee optimality. Something for me to think about.
- Micro optimality
- Generated ticket sets concentrate on improving overall coverage and ignore individual ticket subsets. Rows are sorted alpha numerically, therefore intra-ticket overlap is massive within large sets. Rows could be rearranged to minimise overlap for each ticket lessening differences in coverage between tickets from large and small sets. Reordering rows makes no difference in probability for entire sets. It would however, improve individual ticket odds if used separately.
Frequently Asked Questions
- Why not just distribute a single set of numbers?
- If everyone used the same numbers, they would all win on the same draws and the payout would be lower. By generating a fresh set at each page load, it distributes the winners giving an even spread. It therefore makes sense to generate fresh sets for friends and family. Just hit the generate button.
- Does it matter if I use the first 10 rows of the 30 row set?
- Yes. As rows are sorted, overlap tends to be higher between rows closer together. You are better off using a 10 row set for a single ticket. The same is true the other way around. A 30 row set is superior to 3 independently generated sets of 10. This is because each set of 10 is unaware of the others and therefore total overlap is higher, even though the overlap within each ticket is lower.
- Are these sets optimal?
- Probably not. But they are better than randomly generated ones and those you may choose yourself.
- What about Powerball?
- Because only a single number is drawn, any number is as good as the next. It cannot be improved upon.
- 10 Row Set - 4.986% Chance of a Win
- Ticket 1 of 1
- 01 06 14 26 32 38
01 09 11 18 30 35
02 10 16 31 33 40
03 14 21 27 33 34
04 10 12 24 28 32
05 21 23 24 30 39
05 25 26 29 36 37
06 07 08 19 22 39
08 20 27 28 35 36
13 15 17 20 23 38
318 Visits Since 2007-01-16T10:24:49+13:00 P3Y7M21DT7H7M6S - 42.23ms