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# Theory

As covered in "False Tips", all possible draws for any lottery have the same likelihood of occuring. Therefore, a single row can be expected to perform as well as any other. However, tickets usually have more than one row, and rows that share less winning draws tend to perform better than rows that share more. To explain, we'll use a fictional lottery called "Pot Luck" and do analysis using the Calculator. It has 47 balls, from which 6 are drawn. This lottery has 5 divisions:

Pot Luck's 5 ways to Win!
DivisionMatching Balls1 in Odds
1610,737,573
2543,649
34873
4350
527

This example will be comparing sets of 10 rows each. Here is one of the 10 million plus worst possible sets:

 9 11 17 24 34 44 9 11 17 24 34 44 9 11 17 24 34 44 9 11 17 24 34 44 9 11 17 24 34 44 9 11 17 24 34 44 9 11 17 24 34 44 9 11 17 24 34 44 9 11 17 24 34 44 9 11 17 24 34 44
Worst Possible Results
DivisionCoveragePercentage
11~0
22460.002
312,3000.115
4213,2001.986
51,519,05014.147
Any1,744,79716.249
Any expected wait draws: 6.154

As can be seen above, all Worst's ticket rows are the same. This means it has the same chance of winning as a single row ticket. However, if it does win, it will win on all 10 lines, seemingly making it just as good as any other 10 row ticket. This is not the full truth as:

1. The expected wait is 10 times longer before its first win. For divisions of small odds, this could mean the difference between a life changing win for you, or your distant descendant.
2. Most lotteries, (particularly for the top divisions,) divide the prizes by the number of winners, so winning 10 times on the same division and draw, returns less than winning once on 10 different draws - significantly less if you are the only 1st division winner.

To make the second point more clear, consider a shared prize division that has jackpotted to one million dollars. The table below calculates the expected prizes for winning 10 times on different draws and 10 times on the same draw, with respect to the number of other winners:

10 wins for Shared Prize of \$1M
Other WinnersDifferent DrawsSame Draw
0\$10,000,000\$1,000,000
1\$5,000,000\$909,091
2\$3,333,333\$833,333
3\$2,500,000\$769,230
5\$1,666,667\$666,667
10\$909,090\$500,000
100\$99,010\$90,909
10000\$1000\$999
n10*1e6/(n+1)1e6*10/(n+10)

Thus, when prizes are shared, the expected return for wins across multiple draws will never be less than the expected return for multiple wins on the same draw - often significantly more.

Below we have a slightly better set, where all the rows have one unique number. In terms of coverage, these rows have over 41% probability of some kind of win, while the set above only had 16%. Its numbers are more spread, so it has better coverage.

 1 11 17 24 34 44 2 11 17 24 34 44 3 11 17 24 34 44 4 11 17 24 34 44 5 11 17 24 34 44 6 11 17 24 34 44 7 11 17 24 34 44 8 11 17 24 34 44 9 11 17 24 34 44 10 11 17 24 34 44
Slightly Improved Results
DivisionCoveragePercentage
110~0
21,8670.017
369,5050.647
4874,5008.144
54,365,76040.659
Any4,484,90741.768
Any expected wait draws: 2.394

The set below obviously has greater spread, and its coverage/odds are nearly double the ticket above, nearly halving the expected win wait time. It was generated randomly and shows that randomly generated sets can be pretty good.

 7 21 10 13 39 37 41 47 19 45 13 5 2 4 21 8 25 9 7 17 19 35 32 41 33 23 37 10 25 41 21 9 1 23 17 13 4 29 8 16 25 31 40 19 14 24 8 31 2 11 12 4 34 13 11 19 30 3 21 8
Random Results
DivisionCoveragePercentage
110~0
22,4600.023
3122,3431.140
41,919,49018.172
58,158,89078.245
Any8,583,74679.941
Any expected wait draws: 1.251

The set below is also randomly generated and it is slightly better than the previous. Trouble is, without testing software, it would be extremely difficult to tell that it has an extra 4% probability of winning.

 1 12 32 26 45 6 15 40 43 32 5 16 33 40 24 14 23 35 2 39 16 4 37 15 23 41 5 7 30 27 17 16 30 39 18 2 29 9 1 46 20 40 21 35 29 4 47 6 4 46 12 43 31 27 4 16 43 42 27 3
2nd Random Results
DivisionCoveragePercentage
110~0
22,4600.023
3122,1541.138
41,955,98218.216
58,409,74378.321
Any8,940,25883.261
Any expected wait draws: 1.201

Lastly, here's a ticket generated by an early LT algorithm. It provides yet another 10% of the total coverage, giving the greatest odds and therefore lowest expected wait.

 39 13 9 33 16 46 28 8 17 12 6 32 1 27 36 14 35 24 25 10 20 45 44 42 5 30 41 15 21 36 23 19 3 47 22 34 7 31 38 26 37 40 42 29 21 23 17 27 11 4 18 2 43 29 1 32 5 13 4 44
Results
DivisionCoveragePercentage
110~0
22,4600.023
3123,0001.146
42,071,22819.290
59,130,97685.038
Any10,045,02793.550
Any expected wait draws: 1.069

The average 10 row coverage for this lottery is 83.022%. LT's ticket, at over 93%, is over 10% better. We don't claim LT's numbers are optimal, but we expect them to be significantly better than average. While it's possible that a randomly generated ticket could outperform LT's, it's extremely unlikely. But if you are that lucky, you can test it using the Calculator to verify your results.