# The Naked Scientists Forum

### Author Topic: Impossible lottery combinations  (Read 19186 times)

#### Musicforawhile

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##### Impossible lottery combinations
« on: 07/11/2014 22:09:40 »
If there is an equal chance of any number in the lottery being drawn then why do you never get a combination like 1,2,3,4,5,6 or 9, 12, 15, 18, 21. Even 1,2,3,4,12,36 would probably not happen.

Please don't explain with equations, I am not a mathematician.

#### Bill S

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##### Re: Impossible lottery combinations
« Reply #1 on: 07/11/2014 23:26:05 »
1,2,3,4,5,6 is no less likely than (for example) 7,20,49, 53, 59, 81.  There is something like a 14 million to one chance of each.

#### jeffreyH

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##### Re: Impossible lottery combinations
« Reply #2 on: 07/11/2014 23:44:04 »
Say you have 49 possible selections then one number must occupy one position. You then have 48 possible positions left for the second number and so on until you have accounted for all six selections. That is 49*48*47*46*45*44.
That is a 10,068,347,520 to 1 chance of any selection appearing.

#### RD

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##### Re: Impossible lottery combinations
« Reply #3 on: 08/11/2014 06:26:40 »
all combinations six lottery numbers have the same odds of being picked, even if they were the winning numbers last week ... www.thenakedscientists.com/forum/index.php?topic=26028

see ... http://en.wikipedia.org/wiki/Gambler's_fallacy
« Last Edit: 08/11/2014 06:39:39 by RD »

#### PmbPhy

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##### Re: Impossible lottery combinations
« Reply #4 on: 08/11/2014 06:54:39 »
Quote from: Musicforawhile
If there is an equal chance of any number in the lottery being drawn then why do you never get a combination like 1,2,3,4,5,6 or 9, 12, 15, 18, 21. Even 1,2,3,4,12,36 would probably not happen.
On what are you basing those assumptions on?

Quote from: Musicforawhile
Please don't explain with equations, I am not a mathematician.
Okay. I still don't believe that your assumptions are correct. But remember that those are strings of special characters and are not arbitrary.

#### alancalverd

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##### Re: Impossible lottery combinations
« Reply #5 on: 08/11/2014 08:45:40 »
Lottery is a tax on the poor and stupid.

If you buy a UK lottery ticket, you are more likely to die than win first prize., whatever combination you choose for the next draw.

However you can minimise the damage done to your wealth by your stupidity by not choosing a sequence of numbers, or numbers below 31, because very stuipd people will choose sequences or birthdays, so you will have to share your winnings with lots of poor people, which is against the principles of capitalism. That said, you are now limited to only 13 numbers, so there is a high probability that you will share your winnings with insane mathematicians or people born on another planet, which makes the winners podium much more interesting.

Do, however, keep an eye on the Irish Hospitals Sweepstake.  A few years ago the guaranteed prize money exceeded the total ticket value, so a consortium of Legitimate Businessmen bought the lot.

#### Musicforawhile

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##### Re: Impossible lottery combinations
« Reply #6 on: 08/11/2014 12:20:36 »
I am basing my evidence on the fact that no strings of numbers like that have ever occurred in any lottery anywhere...am
I wrong? PmbPhy said they were special numbers...what does that mean? Can you explain?

And as for the same numbers that came up last week coming up the next week being the same...you've got to be kidding right? Is there an equal chance of those same numbers coming up again week after week?? This is why I hate probability. Real life doesn't work this way.

So let me ask you...what is the probability of 5, 12, 24, 37, 41, 46 coming up in consecutive weeks??? Is it more or less likely to see a completely different set of numbers coming up after the initial drawing of 5, 12, 24, 37, 41, 46?

What is the likelihood that that same set of numbers comes up every week until we finally get hit by a huge asteroid and the planet and our lottery machines are destroyed? You can't tell me its the same likelihood as having a mixed/more different drawing of numbers each week??
« Last Edit: 08/11/2014 12:41:35 by Musicforawhile »

#### dlorde

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##### Re: Impossible lottery combinations
« Reply #7 on: 08/11/2014 13:07:05 »
If there is an equal chance of any number in the lottery being drawn then why do you never get a combination like 1,2,3,4,5,6 or 9, 12, 15, 18, 21. Even 1,2,3,4,12,36 would probably not happen.
Perhaps if you did some research, you'd find your assumptions are questionable.

Maryland Bonus Match 5 lottery, Oct 20, 2004 draw: 33-34-35-36-37, Bonus: 27
Ontario Lottario lottery, May 22, 1999 draw: 1-2-3-4-5-33  Bonus: 25

Interestingly, some people cite this as evidence the lotteries are fixed!

The sad fact is that human intuitions about probability and randomness are extremely unreliable. I hear that Apple had to de-randomise their iPod shuffle function - to make it seem more random...
« Last Edit: 08/11/2014 13:08:44 by dlorde »

#### RD

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##### Re: Impossible lottery combinations
« Reply #8 on: 08/11/2014 14:27:06 »
So let me ask you...what is the probability of 5, 12, 24, 37, 41, 46 coming up in consecutive weeks??? Is it more or less likely to see a completely different set of numbers coming up after the initial drawing of 5, 12, 24, 37, 41, 46?

Any sequence of numbers have the same probability, that some sequences have significance to you , (e.g. sequential numbers or repetition), does not make them more or less likely.

lottery-machines, roulette-wheels and tossed-coins have no memory :
their previous performance does not influence their future performance.
To think otherwise is the gambler's fallacy which I mentioned previously.

#### alancalverd

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##### Re: Impossible lottery combinations
« Reply #9 on: 08/11/2014 17:53:56 »
PmbPhy said they were special numbers...what does that mean? Can you explain?'

Not entirely sure what he meant (nothing new there!) but as I see it: any string of numbers you choose becomes "special" in your mind, but not in the mind of the machine that picks the winners - it simply doesn't know or care about your perception of the world.

Quote
And as for the same numbers that came up last week coming up the next week being the same...you've got to be kidding right? Is there an equal chance of those same numbers coming up again week after week?? This is why I hate probability. Real life doesn't work this way.
Oh yes it does, and it's known as Sod's Law (just kidding). Seriously, if the probability of a random event is p, then the probability of it occurring n times in successive trials is pn. It just happens that in the national lottery, p is very small indeed, so pn = (very)n small. Think of a coin toss. The probability of one head is ½, two successive heads is ¼, and so forth. But it never quite reaches zero.

Quote
So let me ask you...what is the probability of 5, 12, 24, 37, 41, 46 coming up in consecutive weeks??? Is it more or less likely to see a completely different set of numbers coming up after the initial drawing of 5, 12, 24, 37, 41, 46?
The probability of any set of numbers occurring in any one draw is exactly the same because each draw is independent - the machine has no memory.

Quote
What is the likelihood that that same set of numbers comes up every week until we finally get hit by a huge asteroid and the planet and our lottery machines are destroyed? You can't tell me its the same likelihood as having a mixed/more different drawing of numbers each week??
You can calculate it yourself, using the pn formula where n is the number of weeks that will pass before the asteroid impact - generally considered to be about 52 x 108 ± 50%.

[/quote]

#### Musicforawhile

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##### Re: Impossible lottery combinations
« Reply #10 on: 08/11/2014 22:14:26 »
Ok thanks for the replies, even if I don't quite get it yet. And thank you for the asteroid equation

Another thought..if you look at all the lottery draws, more of the winning numbers have no pattern than do have a pattern. The Maryland draw was quite unusual but it's not the norm...so it seems it's more likely that the numbers wont be in a sequence than will be, if you look at all the draws that have happened. So therefore it is more likely that the winning numbers won't have any pattern.

#### jeffreyH

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##### Re: Impossible lottery combinations
« Reply #11 on: 08/11/2014 22:38:17 »
You are fixating on the order that the selection of numbers are presented in as if they mean something. Instead of being the integers 1-49 they could be any unique 49 numbers in no sequential order at all. This is like having a 49 element array X indexed X1 to X49.

#### alancalverd

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##### Re: Impossible lottery combinations
« Reply #12 on: 08/11/2014 23:09:06 »

Another thought..if you look at all the lottery draws, more of the winning numbers have no pattern than do have a pattern.

"Sequence" and "pattern" are your interpretation - and it is something that the human brain is very good at imposing on events which are actually random. My colleagues spent many hours (possibly years) trying to find the cause of "ring artefacts" in CT images - apparent circles in the noise underlying the images. It turned out that the noise was indeed random, and could not have been otherwise as the image is formed by a rectangular scan, not a circular one, but the brain looks for recognisable geometric shapes and "fills in the gaps" wen confronted with white noise.

As far as the lottery machine is concerned, the numbers it spews out are entirely random and independent of one another, but because they are always published in ascending order, people can identify groups that appear to have an arithmetic sequence, or can impose a sequence on a near-sequence.

#### dlorde

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##### Re: Impossible lottery combinations
« Reply #13 on: 09/11/2014 00:16:12 »
Another thought..if you look at all the lottery draws, more of the winning numbers have no pattern than do have a pattern. The Maryland draw was quite unusual but it's not the norm...so it seems it's more likely that the numbers wont be in a sequence than will be, if you look at all the draws that have happened. So therefore it is more likely that the winning numbers won't have any pattern.
That's hardly surprising. If you define a tiny subset of all the possible sequences as interesting (e.g. having a 'pattern'), then naturally you'd expect to come across such a sequence very rarely in a random draw... of course the  Maryland draw isn't the norm.

#### evan_au

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##### Re: Impossible lottery combinations
« Reply #14 on: 09/11/2014 03:17:16 »
For a lottery selecting 6 from 49 balls, you get a
Quote from: jeffreyH
10,068,347,520 to 1 chance of any selection appearing

If you bought one lottery ticket every week with your favourite numbers, that would be around 96,479,716 years before your numbers came up once. That's similar to the interval between major extinction events.

It's a similar wait before the identical numbers come up two weeks in a row (in the same lottery).

You can ignore any odds that your numbers will come up two weeks in a row - it's around 1 in 101,370,000,000,000,000,000. Definitely asteroid time!

Quote from: Musicforawhile
of course the  Maryland draw isn't the norm
I would say that the figures in the Maryland draw are "the normal" in the sense that they are just as likely as any other sequence of numbers.

However, the comment used "the norm" in a colloquial sense of "frequently", which is definitely not normal for a lottery draw!

Quote from: Musicforawhile
Impossible lottery combinations
The impossible combinations are those where the same number comes up more than once, or you get a number higher than 49, or less than 1!

This would require a major malfunction of the lottery machine, which would be declared an invalid draw.

#### RD

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##### Re: Impossible lottery combinations
« Reply #15 on: 09/11/2014 07:05:02 »
You can ignore any odds that your numbers will come up two weeks in a row ...

We should stress the point that that would be true whether the same numbers were chosen each week by the gambler or different ones.

That the numbers won last week does not make it more or less likely they will win this week.
Avoiding/favouring previous winning number selections does not improve ones odds of winning the lottery.

The odds that someone who only buys one lottery ticket per week winning first prize on the UK national lottery on two consecutive weeks is approximately 1 in 200,000,000,000,000  whether they use the same numbers each week or different ones.
« Last Edit: 09/11/2014 07:13:17 by RD »

#### dlorde

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##### Re: Impossible lottery combinations
« Reply #16 on: 09/11/2014 11:07:06 »
You can ignore any odds that your numbers will come up two weeks in a row ...
The odds that someone who only buys one lottery ticket per week winning first prize on the UK national lottery on two consecutive weeks is approximately 1 in 200,000,000,000,000  whether they use the same numbers each week or different ones.
Although the odds of anyone (i.e. not someone specified in advance) winning two weeks in a row would be considerably less (because somebody wins most weeks, and the odds of that person winning the next week are the same as anyone else's - if everybody buys only one ticket).

#### Musicforawhile

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##### Re: Impossible lottery combinations
« Reply #17 on: 09/11/2014 14:22:56 »
You are fixating on the order that the selection of numbers are presented in as if they mean something.  1 to X49.

The numbers could come out of the draw like:

5, 2, 1, 4, 6, 3

The pattern is there once you re-arrange them, unless I'm missing what you're saying.

I think despite all your kind efforts, I just don't have the kind of mind that can accept this sort of thing. In my life, I am involved with art, literature and music and am very much led by my intuition. I do understood that certain things in the universe are counter-intuitive like Einstein's theory where time slows down when you approach the speed of light. Is that right? So I know I can be very wrong in some things, esp. science things.

#### dlorde

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##### Re: Impossible lottery combinations
« Reply #18 on: 09/11/2014 16:00:59 »
The numbers could come out of the draw like:

5, 2, 1, 4, 6, 3

The pattern is there once you re-arrange them, unless I'm missing what you're saying.
Thing is, they're just numbers. You decide what particular combinations are significant to you (you seem to find simple sequences of adjacent numbers significant) and whether they count if they're jumbled (you seem to think not). A mathematician could probably find some kind of pattern in a lot more combinations than you can, and almost every combination could be significant to someone on the planet (birthdays, phone numbers, house numbers, etc).

Quote
I think despite all your kind efforts, I just don't have the kind of mind that can accept this sort of thing. In my life, I am involved with art, literature and music and am very much led by my intuition.
It's easy enough to learn this kind of thing if you have sufficient interest. Unfortunately, although intuition is useful in the arts and socially, it tends to be very unreliable in maths and science - unless you are knowledgeable and experienced in those fields. Intuition is based on rules and heuristics that are, or have become, subconscious, so it can be improved with experience.

Quote
I do understood that certain things in the universe are counter-intuitive like Einstein's theory where time slows down when you approach the speed of light. Is that right?
Time slows down whenever you're moving relative to something you're measuring time on. It's only really noticeable when you're moving at a significant fraction of the speed of light. Some everyday things that move quite fast and require high-precision (e.g. GPS satellites) need to make allowances for it.

#### chiralSPO

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##### Re: Impossible lottery combinations
« Reply #19 on: 09/11/2014 16:11:13 »
This really shouldn't be that confusing, you just have to give it a few minutes of thought. If you have a bowl of scrabble tiles (these aren't evenly distributed, as with numbers in the lottery, but still works) and you pull three letters, most of the time it won't be a three letter word. There are 17576 (263) possible three letter combinations you could draw (in a given order), but the official Scrabble dictionary only lists 1015 accepted 3 letter words. There isn't anything forbidden about randomly drawing a word, it's just less likely than coming up with a combination that isn't a word. Also there is something about our minds that naturally classify the words into recognized real words, possible words, and couldn't possibly be words. I would have lumped "alb" in with "fis" as possible words, but apparently a "alb" is a word, and "fis" is not. Then there are plenty of combinations like LBJ, FDR, JFK, LSD, TKO etc. that are recognizable to some people as acronyms, but would be deemed nonsensical combinations by others.

There are many fewer recognizable lottery numbers than possible combinations of those digits, but the same logic holds.
« Last Edit: 09/11/2014 16:13:04 by chiralSPO »

#### Musicforawhile

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##### Re: Impossible lottery combinations
« Reply #20 on: 09/11/2014 20:10:50 »

Thing is, they're just numbers. You decide what particular combinations are significant to you (you seem to find simple sequences of adjacent numbers significant) and whether they count if they're jumbled (you seem to think not). A mathematician could probably find some kind of pattern in a lot more combinations than you can, and almost every combination could be significant to someone on the planet (birthdays, phone numbers, house numbers, etc).

Yes I was thinking that, that even in something like 7, 16, 21, 33, 39, 42...a mathematician might find a pattern in that.

#### phyti39

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##### Re: Impossible lottery combinations
« Reply #21 on: 09/11/2014 20:21:28 »
The lottery system has no memory, therefore each drawing is like the first ever. This eliminates any causal connection, and enforces a fair system. As mentioned, any sequence has the same probability as any other. Just as the toss of a coin is time independent, so is the lottery.
When someone says multiple occurrences of the same number are rare, they are in reality saying it cannot happen in there lifetime. This is false based on time independent behavior, it can occur anytime.

#### Musicforawhile

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##### Re: Impossible lottery combinations
« Reply #22 on: 09/11/2014 20:21:43 »
Ok, if someone said they would buy you a lottery ticket for the next ten years, one each week. Would you prefer to choose a set of numbers that:
2. Were randomly generated each week
3. You really wouldn't care whether it was 1 or 2 as the probability is the same (not because you just don't care about the lottery)

This is assuming you can imagine caring about the tiny chance you have of winning the lottery.

#### dlorde

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##### Re: Impossible lottery combinations
« Reply #23 on: 09/11/2014 20:42:06 »
Ok, if someone said they would buy you a lottery ticket for the next ten years, one each week. Would you prefer to choose a set of numbers that:
2. Were randomly generated each week
3. You really wouldn't care whether it was 1 or 2 as the probability is the same (not because you just don't care about the lottery)
3 - it makes no difference whether the numbers have come up before or are randomly generated. Intuitively, one feels it should make a difference, just as one feels that after tossing a coin 10 heads in a row you're more likely to get tails. Intuition is wrong about this.

As already mentioned, the only good reason to select particular numbers is to avoid popular choices, so that if you do win, you're less likely to have to share the winnings with a lot of people.

#### Musicforawhile

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##### Re: Impossible lottery combinations
« Reply #24 on: 09/11/2014 21:25:56 »

3 - it makes no difference whether the numbers have come up before or are randomly generated. Intuitively, one feels it should make a difference, just as one feels that after tossing a coin 10 heads in a row you're more likely to get tails. Intuition is wrong about this.

My mind can't accept this, not at the moment anyway. And I think that if all the mathematicians here weren't under the gaze of their colleagues, academic supervisors, and potential academic funders then you would all go for Number 2. But you don't want to be seen as woolly headed and led by your instinct rather than your rationality. In the same way I am sure many scientists wouldn't want to admit their secret agnosticism or prayers to god.

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##### Re: Impossible lottery combinations
« Reply #24 on: 09/11/2014 21:25:56 »