1,2,3,4,5,6 in the lottery

[Skeptic Ginger]

I think what jp means is that a combination of patterned numbers is less likely than a combination of un-patterned numbers, which, of course, is correct. He just didn't word it very well.



"No" to the first/second sentence or third? If the first, he's correct, and certainly does, seemingly, understand probability theory, at least to that extent.

What is the principle for the patterned and un-patterned combos differing in odds? How is any single 6 number pick any more or less likely than any other single 6 number pick?

 

[psionl0]

What is the principle for the patterned and un-patterned combos differing in odds? How is any single 6 number pick any more or less likely than any other single 6 number pick?

It isn't.

However, if you choose from the same 9 or 10 numbers to fill in all your tickets and the draw includes the same numbers then more of your tickets will win as compared to filling your tickets from random numbers. (OTOH if the draw doesn't include those numbers then none of the tickets win).

In the long run, the expected value of your winnings (or rather, your losses) will be the same no matter what pattern you use to fill in your tickets.

 

[Dan O.]

Guy buys $1 ticket and loses, but can deduct $1 from (future) winnings, so lost $1 but will save 35 cents in future taxes, net loss 65 cents.

Is this true in New Zealand? The tax code in the US does not allow gambling losses to be carried forward to later years.

 

[Dan O.]

I see that as being one of the “other advantages”. If there are multiple winners of First Division the prize money is shared equally between all winning tickets. As you hold ten winning tickets you get ten shares instead of only one (I don‘t care how “messy” that is).

I thought you were better at maths.

Another approach is to simultaneously play every possible game and see which method comes out ahead. There are about 4 million combination for the regular lotto and 10 combinations for the power ball. In each game we are playing 10 tickets. For $1 tickets, we each need to start with $400 million. In each game, matching the regular lotto will pay $1 million divided equally to each winning ticket. Matching the regular lotto with a matching power ball will pay an additional 30 million.

In each game, I will play my 10 tickets on separate numbers:

Ticket 1 = 1,2,3,4,5,6 P1
Ticket 2 = 4,5,6,7,8,9 P2
Ticket 3 = 7,8,9,10,11,12 P3
Ticket 4 = 10,11,12,13,14,15 P4
Ticket 5 = 13,14,15,16,17,18 P5
Ticket 6 = 16,17,18,19,20,21 P6
Ticket 7 = 19,20,21,22,23,24 P7
Ticket 8 = 22,23,24,25,26,27 P8
Ticket 9 = 25,26,27,28,29,30 P9
Ticket 10 = 28,29,30,31,32,33 P10

Each of my tickets will win the lotto jackpot in 10 separate games and the power ball jackpot in 1 game. I will get back $1 Million times 10 tickets times 10 games plus $30 Million times 10 tickets times 1 game for a total of $400 Million.

Now, lets see how you will do. In each game, you buy 10 tickets with all the same lotto numbers and a different power ball number.


Ticket 1 = 1,2,3,4,5,6 P1
Ticket 2 = 1,2,3,4,5,6 P2
Ticket 3 = 1,2,3,4,5,6 P3
Ticket 4 = 1,2,3,4,5,6 P4
Ticket 5 = 1,2,3,4,5,6 P5
Ticket 6 = 1,2,3,4,5,6 P6
Ticket 7 = 1,2,3,4,5,6 P7
Ticket 8 = 1,2,3,4,5,6 P8
Ticket 9 = 1,2,3,4,5,6 P9
Ticket 10 = 1,2,3,4,5,6 P10

Each of your tickets will also win the lotto jackpot in 10 games but since they all win in the same games, the prize each ticket wins is divided 10 ways. Each of your tickets also win a power ball jackpot in one game just like mine. You get back $1 Million split 10 ways times 10 tickets times 10 games plus 30 million times 10 tickets times 1 game for a total or $310 Million.

 

[Brian-M]

Buying two tickets in the Big Wednesday doubles the chances (attempts at winning) the first draw but it doesn’t halve the odds. The winning odds of the first ticket are virtually the same as for the second. Covering the second coin toss draw with the two tickets changes the odds from being 1 in 2 to being a guaranteed win.

To make things simple, let's assume that the chances of getting all six numbers right are one chance in 10 million, and the coin-toss reduces your chances of any given ticket winning to one chance in 20 million.

Buying two tickets with the same six numbers, but a different heads/tails option would mean that together those two tickets give you a one in 10 million chance of winning the main prize, because the coin toss is now irrelevant.

But... if buying a ticket gives you a one in 20 million chance of winning the main prize, then buying two tickets with different numbers would double your chances of this happening, giving you a one in 10 million chance of winning the main prize.

In both cases, the chances of winning the main prize is the same, with the same number of tickets purchased. There is no advantage either way.

ETA:

The winning odds of the first ticket are virtually the same as for the second.

And since you have two tickets this doubles your chances of winning, compared to having only a single ticket.

Covering the second coin toss draw with the two tickets changes the odds from being 1 in 2 to being a guaranteed win.

In other words, this doubles your chances of winning, from 1 in 2 to 2 in 2, if you get the right six numbers, but halves you chances of getting the right six numbers compared to buying the same number of tickets with different numbers.

If you double your chances and halve them, nothing has changed.

 

[alexi_drago]

What is the principle for the patterned and un-patterned combos differing in odds?

Consider 6 consectutive numbers, in a 6/49 lottery there's 44 combinations, then if they increase by 2 instead of 1 each time there's 39 combinations. Try to think of other ways that 6 numbers out of 1-49 could form some sort of obvious pattern. You could come up with a few thousand maybe, but that's out of 13983816 combinations and the remainder of those have a more random appearance.
Say you could see some sort of patterns in 10000 different cominations then you'd have:

10000 patterned
13973816 un-patterned

so the chances of seeing some sort of pattern drawn are lower than 1 in 1000

How is any single 6 number pick any more or less likely than any other single 6 number pick?

No single 6 number pick is any more likely than any other, it's just that there are many more un-patterned combos so it's more likely to be one of them.

Any 6 numbers you choose have the same odds as 1,2,3,4,5,6 of coming up but since an obvious pattern is rarely seen (less than 1 in a thousand draws if you take my example above) then some people think they have a better chance by choosing a more random looking combo.

 

[Brian-M]

How many of you here would play a "fair" lotto with "good" odds, if one existed?

(Having fun making up an example...)

Let's say you have a lotto called Fortune Four, where you pick four numbers from 64. Tickets cost $4 (plus newsagent or paypal fee). If you get all four numbers right, you win $1,500,000. If you get three numbers right, you win $13,000.

In this case, the lotto would be returning 90% of the money it receives to the players, and players would have a 1 in 635,376 of winning the $1,500,000.

Probably the only public lotto to give better than one in a million chance to win more than a million dollars.

Someone who played 4 games a week would have a better than 1 in 100 chance of winning the $1,500,000 within 31 years.

Would you consider this to be a lotto worthwhile playing, or still a waste of money?

(I hope I worked out all the figures right. No doubt someone will correct me if I haven't.)

 

[alexi_drago]

How many of you here would play a "fair" lotto with "good" odds, if one existed?

(Having fun making up an example...)

Let's say you have a lotto called Fortune Four, where you pick four numbers from 64. Tickets cost $4 (plus newsagent or paypal fee). If you get all four numbers right, you win $1,500,000. If you get three numbers right, you win $13,000.

In this case, the lotto would be returning 90% of the money it receives to the players, and players would have a 1 in 635,376 of winning the $1,500,000.

Probably the only public lotto to give better than one in a million chance to win more than a million dollars.

Someone who played 4 games a week would have a better than 1 in 100 chance of winning the $1,500,000 within 31 years.

Would you consider this to be a lotto worthwhile playing, or still a waste of money?

(I hope I worked out all the figures right. No doubt someone will correct me if I haven't.)

if I bought 635,376 tickets I would be guranteed to get over 4.5M back. Yes, I'd play.

 

[DazzaD]

if I bought 635,376 tickets I would be guranteed to get over 4.5M back. Yes, I'd play.

Assuming noone else plays the game and shares the prize with you?

 

[alexi_drago]

Assuming noone else plays the game and shares the prize with you?

If you get all four numbers right, you win $1,500,000. If you get three numbers right, you win $13,000.

He didn't mention sharing.

 

[zooterkin]

For years, I've been telling people that it's as likely for you to get the above numbers as something like 35,32,18,16, 28, 21.

People can't seem to believe it. Is this actually the case, or am I making a fool of myself?

You're correct (as others have said already), and you're also not alone, as the exact same question was addressed on the excellent More or Less. The podcast version is here; the January 6 edition has the item, and there's a follow-up in the Jan 13th one (they had the correct explanation, but they got the announcement wrong in the illustration they did; they got the regular lottery announcer to read out the numbers, but tacked on "in that order", which does change the probabilities involved).

 

[Brian-M]

if I bought 635,376 tickets I would be guranteed to get over 4.5M back. Yes, I'd play.

Ah, I guess I did make a mistake with the figures. I worked out the wrong number of minor prizes possible, and therefore the wrong return for them.

According to a different quick calculation (which is probably also wrong because I'm distracted by watching House at the moment) shows that the minor prize should only be about $88 ("about" because I'm rounding it up to a more aesthetically pleasing value).

ETA: No, that is wrong too. I'll figure it out later.

ETAA: Let's see...

Four numbers drawn from 64 leaves 60 incorrect numbers. So out of the 635,376 tickets, 60 will have only the first number wrong, 60 will have only the second number wrong, 60 will have only the third number wrong, and 60 will have only the last number wrong.

That's 240 minor prizes.

(($4 * 635,376 * 0.9)-$1,500,000)/240 = $3,280.64

So the minor prize should be $3,300 instead of $13,000.

 

[vladi]

I would imagine your best chance of not having to share the payout if you do win is to pick whatever numbers came up the previous week. They have exactly the same chance of coming up this week as any other combination, but you'll likely to be the only person playing them.

Not so. This actually happened here in Bulgaria: www dot dailyfinance dot com/2009/09/18/bulgaria-lottery-repeats-winning-numbers-and-winners-abound/
And guess what? 18 people won the second time!

 

[Ladewig]

Not so. This actually happened here in Bulgaria: www dot dailyfinance dot com/2009/09/18/bulgaria-lottery-repeats-winning-numbers-and-winners-abound/
And guess what? 18 people won the second time!

Each one won around 7600 $US.

The lottery chooses six number from 1 to 42 without replacement. The odds of hitting all 6 are around 1 in 5.2 million. If the grand prize (before the split) was around 140,000 $US than the lottery is pretty much a suckers' game designed to separate rubes from their money.

Vladi's link:
http://www.dailyfinance.com/2009/09/18/bulgaria-lottery-repeats-winning-numbers-and-winners-abound/

That story cites odds of 1 in 4.2 million. That probability was calculated by a mathematician who assumed that small flaws in the balls cause the same numbers to be repeated more often that probability would predict. Also, some stories cite $7300 and others cite $7643.

 

[Almo]

And just because it happened once doesn't prove the probability is not astronomically small.

 

[vladi]

Yes, but it proves that there are enough people that play with last week's numbers and you are not very likely to be the only one that wins in such occasion Some people may consider them nuts but... they won

 

[Ladewig]

Yes, but it proves that there are enough people that play with last week's numbers and you are not very likely to be the only one that wins in such occasion Some people may consider them nuts but... they won

Yes. They won.
They won one-eighteenth what the average jackpot winner would win. They can each buy a somewhat crappy used car.
ETA: unnecessary snarkiness removed.
[I wonder if there is a record for tiniest jackpot]

The other thing that sucks is that usually when one has a one-in-five-million story to tell, one can at least get some free drink in the pub for telling a good story. But with this story, others will expect the teller to buy the drinks because he won the lottery.

 

[jasonpatterson]

No.

With a clear and meaningful refutation of my point like this, I hardly need to ask for clarification, but in what way am I wrong, pray tell?

I know it's hard to actually read what another person wrote, but unless you believe that there are an equal or greater number of patterned sets than non-patterned sets in all of the possible lottery outcomes, I can't see why you'd respond in the negative to the first.

Regarding the second clever response, please explain how a lottery wheel, a tool for generating combinations, can change the statistics of the game itself? Whether I choose a random set of numbers or allow a wheel to generate the numbers for me, in a fair game, either ticket is equally likely to win.

 

[dlorde]

I know it's hard to actually read what another person wrote, but unless you believe that there are an equal or greater number of patterned sets than non-patterned sets in all of the possible lottery outcomes, I can't see why you'd respond in the negative to the first.

Perhaps it depends what you mean by a 'patterned set'?

For some reason this reminds me of the 'smallest uninteresting number' paradox (the smallest uninteresting number is interesting for that very reason).

 

[jasonpatterson]

The lottery chooses six number from 1 to 42 without replacement. The odds of hitting all 6 are around 1 in 5.2 million. If the grand prize (before the split) was around 140,000 $US than the lottery is pretty much a suckers' game designed to separate rubes from their money.

I'm fairly sure that most gambling (not all, but most) is a suckers' game designed to separate rubes from their money. That aside, do we know how much a Bulgarian lottery ticket costs? That's going to make a big impact on how bad the game is slanted against players. If you can get a ticket for $0.05 then the prize isn't so bad. If it's $1 per ticket, then wow. Just wow.