Lottery Tickets

[Jon Corby]

Somebody on another forum just commented that the fewest number of lottery tickets you can buy to guarantee £10 is 163. I thought about whether or not this was true, and then how to go about making such a selection of tickets that guarantees that you match 3 of the 6 numbers. I've given it a teensy bit of thought, and besides starting with 1,2,3,4,5,6 as my first ticket (since it can be anything!) I don't know what to do next. (It's an impressive start, I know.)

Any ideas?

[Jon Corby]

It's okay everyone, I found some stuff here about it. Looks like there is a challenge on to try and reduce it, so I guess it hasn't necessarily been proven to be the optimum, but it's probably thereabouts.

Good discussion though. THREAD OF THE YEAR.

[Gavin Chipper]

Thread of the year already. Definitely.

Edit - OK, how about having 7,8,9,10,11,12 as your second ticket?

[Gavin Chipper]

The probability of winning at least £10 with one ticket is about 1/54 (it's 1 in 53 point something - call it 1/n). For each extra ticket you buy, you will add at most another 1/n so you need at least n tickets even before you consider that things work against you when you start using numbers twice.

So 1-6, 7-12, 13-18, 19-24, 25-30, 31-36, 37-42, 34-48 gives you 8 tickets and because they all use different numbers I think we should be up to 8/n already. Then I think you want to equalise as much as possible the number of times each number is used in your selections (call it a guess) while also minimising the amount numbers are used together. So after 43-48, you'd definitely want 49 in your next ticket and then five others that haven't been used together and there may be a neat way of working out what this adds to the 8/n. So you might say 1,7,12,19,25,49. Now every number has been selected once or twice and you'd always use numbers that have been used fewest for your next selection. So maybe 2,8,13,20,26,32. Anyway, I'll come back to this later.

I haven't read any of the stuff on the other site by the way.

[James Robinson]

So 1-6, 7-12, 13-18, 19-24, 25-30, 31-36, 37-42, 34-48 gives you 8 tickets and because they all use different numbers I think we should be up to 8/n already.

Now that one should give you a very strong chance of winning the lottery

[Jon Corby]

Thread of the year already. Definitely.

Edit - OK, how about having 7,8,9,10,11,12 as your second ticket?

That was my immediate thought, but then I also reasoned that maybe you don't need to even cover every number - you obviously don't need to cover anywhere near all combinations as each draw will throw up 20, and you only need to have one of them. (and obviously having some combos will mean you don't have to bother about others)

That page I linked to doesn't really discuss any kind of methodology - it just lists a kind of top 10 of submitted answers, and details what is presumably the best one - plus a challenge to try and beat it. All of which suggests that there isn't (yet?) a definitive answer, nor therefore a way to work it out. Maybe. I mean, the page could be out of date for all I know, it's only a text file.

[Gavin Chipper]

Thread of the year already. Definitely.

Edit - OK, how about having 7,8,9,10,11,12 as your second ticket?

That was my immediate thought, but then I also reasoned that maybe you don't need to even cover every number - you obviously don't need to cover anywhere near all combinations as each draw will throw up 20, and you only need to have one of them. (and obviously having some combos will mean you don't have to bother about others)

I think it is more efficient to use all the numbers because I think every time you introduce a new ticket, you are covering more options by using the numbers you've used least so far. But I think I was wrong that by having 7-12 as your second set doubles your chances because there would be some combinations where 1-6 and 7-12 would both win you £10. So I think the overlap means you would slightly less than double your chances.

That page I linked to doesn't really discuss any kind of methodology - it just lists a kind of top 10 of submitted answers, and details what is presumably the best one - plus a challenge to try and beat it. All of which suggests that there isn't (yet?) a definitive answer, nor therefore a way to work it out. Maybe. I mean, the page could be out of date for all I know, it's only a text file.

Do they explain how they prove a set of tickets works? I think my way would be to simply make a list of tickets using a system based on what I've said and then somehow work out where I can stop.

[Jon Corby]

Do they explain how they prove a set of tickets works? I think my way would be to simply make a list of tickets using a system based on what I've said and then somehow work out where I can stop.

Click it already!

But no, it doesn't explain anything particularly. You can submit your set of tickets to be 'tested', it seems there is a program which validates entries and also details the number of combinations covered. It's only about 3,000.

I'm wondering if the best thing to do is strip it down to a smaller lottery of say 3 balls being drawn from 6 and getting a prize for 2 matches, and seeing what kind of 'pattern' (if any) you end up with that, as that should probably be reasonably easy to optimise. Maybe. I dunno, I'm too busy at work at the moment to give it much thought. I plan to at some point though...

[Howard Somerset]

I'm wondering if the best thing to do is strip it down to a smaller lottery of say 3 balls being drawn from 6 and getting a prize for 2 matches, and seeing what kind of 'pattern' (if any) you end up with that, as that should probably be reasonably easy to optimise. Maybe. I dunno, I'm too busy at work at the moment to give it much thought. I plan to at some point though...

Looks a good approach. And I might try it over the weekend. Maybe start even lower, for example. 3 from 5 with a prize for 2. Anything lower than that would be trivial.

[Liam Tiernan]

Reminds me of this guy http://en.wikipedia.org/wiki/Lottery_Wh ... Full_Wheel who tried to buy up every possible combination for one rollover jackpot in the Irish national lottery. The scheme only worked because although the jackpot itself was split 3 ways, a special Bank Holiday weekend promotion also guaranteed £100 to each match 4 winner, where the usual payout was an average £20, enough to put them into profit. After deducting the costs of setting up the syndicate, paying people to buy batches of tickets, etc, it was estimated that the total return on investment was about 15%.

[Gavin Chipper]

Click it already!

I was worried it might be too spoilerific, but if it doesn't mention methodology I might look then.

You can submit your set of tickets to be 'tested', it seems there is a program which validates entries and also details the number of combinations covered. It's only about 3,000.

If the current best is 163, I might make a list of tickets in the way I think would work best and just stop at 162 and submit that.

[Gavin Chipper]

It seems that there's quite a few on 163 so it's likely that it's the optimum as I'm sure people have worked out the best way to do this. I was thinking a reasonable way of generating selections might be this:

1. Determine how may times each number has been used so far.
2. Pick the number currently used the least (or randomly from the tied selection). This is the first number.
3. For each next number in the selection look at all the other numbers and find one that best fits the priority list, which is:

1. Minimise the number of sixsomes that have been already been used before.
2. Minimise the number of repeated fivesomes.
3. Minimise the number of repeated foursomes.
4. Minimise the number of repeated triplets.
5. Minimise the number of repeated pairs.
6. Minimise the use differential between numbers (use the least used number).

I originally wrote it the following way, but I'm not sure it's defined as well. The priority list:

1. A number that has not been used in the same selection as any of the other numbers currently in the list.
2. A number that has been paired with just one
3. A number that has been paired with two, but never as a triplet in the same selection
4. Paired with three but never as triplet or more
5. Four but never triplet or more
6. Five but never triplet or more
7. Two as a triplet
8. Three with triplet but no more
9. Four with triplet, no more
10. Five with triplet, no more
11. Three as foursome
12. Four with foursome, no more
13. Five with foursome, no more
14. Four with fivesome
15. Five with fivesome, no more
16. Five with sixsome

If there is a tie, always go with a number that has been used less, and if there is still a tie, pick at random.

Then start again with a new selection. Repeat until you have 163 selections. See if that works. Then chop of the last selection and see if 162 works. Corby, you're a programmer aren't you?

[Gavin Chipper]

Obviously I don't know if the above is the best way of going about things, or whether by thinking of purely the next number you do damage to your later moves, but I can't think of any obvious reason why it should.

Also it might be worth running the program a few times to see if doubles and any triples always appear at exactly the same point in the process and so exactly the same number of them come up each time.

[Gavin Chipper]

Did you give up with this, Corby?

[Ryan Taylor]

I won £23.50 on the Euromillions yesterday. Someone do some probability for that...Dmitry...Dmitry...your lunch is ready.

[David Williams]

Reminds me of this guy http://en.wikipedia.org/wiki/Lottery_Wh ... Full_Wheel who tried to buy up every possible combination for one rollover jackpot in the Irish national lottery. The scheme only worked because although the jackpot itself was split 3 ways, a special Bank Holiday weekend promotion also guaranteed £100 to each match 4 winner, where the usual payout was an average £20, enough to put them into profit. After deducting the costs of setting up the syndicate, paying people to buy batches of tickets, etc, it was estimated that the total return on investment was about 15%.

In the days when football pools were big business, and before they started the pools panel, someone did the equivalent. I think the rules were that postponed games counted as away wins, so long as at least 30 games were played. So if you were aiming to get eight draws a bet on a postponed game was money wasted. One day only 31 games survived the weather, and someone with a radio to his ear turned up at Littlewoods at ten to three with a large cheque and a coupon with a perm of any eight from 31. The sort of thing you wish you'd thought of first.

[JimBentley]

In the days when football pools were big business, and before they started the pools panel, someone did the equivalent. I think the rules were that postponed games counted as away wins, so long as at least 30 games were played. So if you were aiming to get eight draws a bet on a postponed game was money wasted. One day only 31 games survived the weather, and someone with a radio to his ear turned up at Littlewoods at ten to three with a large cheque and a coupon with a perm of any eight from 31. The sort of thing you wish you'd thought of first.

I like the story, but a perm of 8 from 31 is 7,888,725 lines and assuming the guy wasn't already loaded, he could only have staked something like 0.01 pence per line or whatever (still £788), so how high would his expected winnings have been? Would they necessarily be more than the stake? Actually I'll shut up now because typing this has made me realise that I'm a bit hazy on how pools payouts used to work.

[David Williams]

My recollection is a bit hazy, but I think he did put a fair sum in. He must have been planning it for years, and would need the number of fixtures to be very close to 30. Every additional game means a lot more stake with no increased return. I would have thought he'd expect a good return, though. If nearly half the games were postponed, only a very small percentage of random selections would not contain at least one of them.

[Jon Corby]

Did you give up with this, Corby?

Oh yeah, I just forgot about it totally. I think I kinda lost interest when I saw the solutions and there didn't seem to be a clear pattern or anything, which was compounded by the fact that they weren't really sure if it was the optimum. I'm kinda of the mindset that I'm never gonna be able to work out something that other people can't.

[Gavin Chipper]

Did you give up with this, Corby?

Oh yeah, I just forgot about it totally. I think I kinda lost interest when I saw the solutions and there didn't seem to be a clear pattern or anything, which was compounded by the fact that they weren't really sure if it was the optimum. I'm kinda of the mindset that I'm never gonna be able to work out something that other people can't.

Yeah you can. You can start by writing program to implement my method.

[Dmitry Goretsky]

I won £23.50 on the Euromillions yesterday. Someone do some probability for that...Dmitry...Dmitry...your lunch is ready.

I'm here

[Ian Volante]

I won £23.50 on the Euromillions yesterday. Someone do some probability for that...Dmitry...Dmitry...your lunch is ready.

I'm here

Kirk