Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Musicforawhile on 07/11/2014 22:09:40
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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.
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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.
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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.
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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 (http://en.wikipedia.org/wiki/Gambler%27s_fallacy)
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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?
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.
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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.
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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??
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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 (http://www.dailymail.co.uk/home/moslive/article-1334712/Humans-concept-randomness-hard-understand.html)...
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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 (http://en.wikipedia.org/wiki/Gambler%27s_fallacy) which I mentioned previously.
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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.
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.
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.
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%.
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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.
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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.
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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.
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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.
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For a lottery selecting 6 from 49 balls, you get a
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!
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!
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.
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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.
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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).
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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.
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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).
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.
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.
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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.
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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.
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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.
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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:
1. Had already been the winning numbers
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.
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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:
1. Had already been the winning numbers
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.
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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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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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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.
As I said, one intuitively feels that runs in random sequences should change the subsequent probabilities, but this obviously isn't possible if each draw or toss is random. Making this mistake is known as the Gambler's Fallacy because it has tempted so many gamblers to their doom. Once you're aware that your intuition has problems with randomness, you learn to use the maths and ignore your intuition (mathematicians do that anyway). It's not a question of who is watching you, it's a question of making the correct decisions and calculations, and getting the right results.
Runs that seem improbable (e.g. ten consecutive heads or tails) don't get 'corrected' in subsequent tosses, they just become insignificant in the long term (thousands of tosses), so the average result remains about 50:50. This is known as 'regression to the mean'.
It may be of interest that in a number of sports (e.g. basketball, baseball, and others), when individual or team performances have been analysed, 'hot' streaks, when a player or team is said to be 'on form' and scoring consistently (or when they're off-form, having a bad spell), the sequences are often indistinguishable from random variation - that is, the results are often consistent with what you should expect from the random 'luck' of a player or team playing at a consistent level of performance rather than their playing form going up or down.
The same kinds of effects are also seen in other areas, such as in finance with the successes or failures of investors and fund managers, and so-on. It pays to be aware of such possibilities.
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I am not a professional physicist, mathematician or otherwise under anyones supervision. It's random and number 3. Not because I guard a reputation but because that is the truth. You can tilt at the windmills all you want. The truth remains the truth.
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the odds of you winning the lotto are 1 in 14 million ( 49!/(6!*(49-6)!) )
so far there have been 1970 lotto draws so you still have 14 million minus 1970 draws to go before the probability of your 1,2,3,4,5,6 selection appearing once has been reached and even then I think its 50:50 whether it would. There simply have been no where near enough draws for it to be likely that any specific selection of 6 numbers will have appeared. The odds are simply way to small and will remain so until long after you have turned your toes up and your bones have decayed to dust. So don't hold your breath while you're waiting for your numbers to come up.
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I am involved with art, literature and music and am very much led by my intuition
Let's say we take a keyboard which has about 49 characters (let's say 24 UPPER and lower case, plus space) and then let a child pound on the keyboard.
If they could type a random sequence of 6 characters every 10 seconds, how long until they type the first 6 letters of "Romeo and Juliet"? (even if you ignored the order in which the letters were typed...)
Answer: There are better things to do with your life!
The odds are similar to winning the lottery, only you have one attempt every 10 seconds instead of every week (and you don't have to pay a few pounds or dollars every time!).
Writing a comprehensible story is hard, which is why we don't teach kids the "random bashing on the keyboard" method for producing great literature, no matter how much your intuition tells you that it should produce text that follows a pattern.
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Writing a comprehensible story is hard, which is why we don't teach kids the "random bashing on the keyboard" method for producing great literature, no matter how much your intuition tells you that it should produce text that follows a pattern.
Probability says that if you give enough monkeys a typewriter each and enough time, one of them will eventually reproduce the complete works of Shakespeare. This probability theory has been disproved by virtue of the fact that the internet has failed to produced anything coming anywhere near close.
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The ability of the human mind to see patterns, or impose apparent order on actual chaos, is amazing. Recognisable images of Jesus and Mohammed appear on slices of toast or in clouds, despite the fact that no living person has seen either bloke in the flesh or even in a contemporary portrait. Listening to Glass or Stockhausen, people often say that they can hear music, and the Raudive Tapes seemed to capture the voices of Hitler or St Francis of Assisi, depending on the listener's preference.
So it would come as no surprise if a monkey won the Booker Prize. It could hardly be more tedious than Anita Brookner, or less meaningful than most of the posts about "consciousness" on this board!
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I hate speculating about lottery numbers, it would be heart breaking if one wrote down a series of numbers for some reason or other, did not buy a ticket and then found you had written down the winning sequence although normally you would be better off by the price of a ticket.
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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.
If you're not going to believe us if we say "3," then what is the point of asking the question in the first place?
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If you're not going to believe us if we say "3," then what is the point of asking the question in the first place?
True, it was a bit pointless of me to ask. The only answer I truly would accept is that you would think "3" but do "2."
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The only answer I truly would accept is that you would think "3" but do "2."
You'd make a lousy gambler. I strongly suggest you stay away from games of chance where money is involved.
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There have in fact been a few people who have won major lotteries twice, or even more, although many of the multiple wins are smaller lotteries with under $1 Million in prize money.
http://abcnews.go.com/blogs/business/2013/02/arkansas-couple-wins-lottery-twice-in-one-weekend/
http://www.lotterypost.com/tag/Won_multiple_times
https://www.thelotter.com/blog/2014/10/twice-three-times-lottery-winner/
Most of those individuals winning multiple times also spend A LOT OF MONEY on tickets.
As far as whether there is a bias towards certain numbers, if the system is truly random, then there should not be.
For something like selecting balls from a cage. Perhaps even a few milligrams difference in the weight of the balls, or a few cubic millimeters in volume would make a difference. Or, perhaps the lottery commission has a verification phase in which all balls are ordered before dumping into the cage, and for some reason they don't get fully mixed.
Thus, there are people trying to calculate the statistics for the perfect card. But, the differences SHOULD BE extremely minimal. Physical characteristics of the balls might also change over time, so data from 10 years ago may not be applicable today.
It is best to consider a lottery as a tax. If the payout is 50%, then if you invest $1,000,000, you should expect to get less than $500,000 back. But, it is worse than that. If the payments are spread out over time, then the lost investment income should be significant. And, taxes also take a chunk out of it.
So, your $1,000,000 investment starts looking a lot like a $100,000 return.
So, can you "beat the odds"? If there is a long streak of no winners, the lotteries often reach a point where the expected payout is greater than the odds. So, for example, it reaches a point where the odds are 1:1,000,000 to win $2,000,000. Outwardly the odds look favorable. But, the payouts are often greater than what is needed. But, also, considering time amortization, and taxes, the actual break even point may be much higher.
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There have in fact been a few people who have won major lotteries twice, or even more, although many of the multiple wins are smaller lotteries with under $1 Million in prize money.
Wouldn't it be true that the occasional multiple winner supports the idea of true randomness?
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For something like selecting balls from a cage. Perhaps even a few milligrams difference in the weight of the balls, or a few cubic millimeters in volume would make a difference. Or, perhaps the lottery commission has a verification phase in which all balls are ordered before dumping into the cage, and for some reason they don't get fully mixed.
In a mindboggling waste of public resources, the UK lottery balls are weighed, measured and assessed for sphericity by the National Physical Laboratory, and the results are analysed for randomness by the Office for National Statistics. Despite all of which, number 20 seems to be an outlier.
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In a mindboggling waste of public resources, the UK lottery balls are weighed, measured and assessed for sphericity by the National Physical Laboratory, and the results are analysed for randomness by the Office for National Statistics. Despite all of which, number 20 seems to be an outlier.
Outliers are not unexpected in random selections; once a 'run' occurs, regression to the mean can take quite a while, so the outlier persists longer than intuitively expected.
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I wonder if lotteries would be more profitable if the government would actually try to convince people that there was a way to beat the odds rather than expending a great effort to convince them that it is random.
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If there is a long streak of no winners, the lotteries often reach a point where the expected payout is greater than the odds.
In cases like this, the institutional gamblers come in.
They pay thousands of students to fill out lottery cards with different ranges of numbers, and lodge them with ticket sellers.
The odds are such that if their system is the only entry with the winning numbers, they will make a profit. But if they have to share the prize with another winner (perhaps a small gambler, or another institutional gambler), then they will make a loss. Regardless of whether the winner makes a profit, the lottery operator still makes a profit.
Unfortunately, the practicalities of filling out forms and lodging them means that some numbers may not be filled in (or filled in twice), or may not make it to the ticket seller by the deadline. It is in the interests of the lottery operator not to make it too easy to take out the whole lottery, thus giving the small gamblers a chance - and give them the illusion that if they buy more tickets they have a chance to win big.
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I appologise if I have posted this in the past, but my interest in infinity demands an infinite lottery. [:D]
In an infinite universe, an infinite lottery becomes possible, and therefore inevitable, not only that, it must occur an infinite number of times. So, what would this infinite lottery be like? There would be an infinite number of people taking part, the staked money would be infinite, therefore, the jackpot (being a percentage of the stake) would also be infinite, the jackpot winners (being a percentage of the infinite number of people taking part) would be infinite, as would the number of losers. We can see from this that an infinite number of people would win an infinite share of an infinite amount of money, but, paradoxically, the same infinite number of people would not be winners at all.
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If the lottery pays out at a finite rate or in a finite period though, these paradoxes can be eliminated. A million dollar lottery that draws every week would have a finite pool of tickets and ticket holders as well as finite buy-in and payout for each cycle, so even though the process repeats infinitely, the infinity isn't involved in any one game.
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In an infinite universe, an infinite lottery becomes possible, and therefore inevitable...
I doubt that an infinite lottery is possible whether the universe is infinite or not (except in the sense that chiralSPO suggests).
I'm also not sure whether everything possible must necessarily happen in an infinite universe - I think it depends on the event and what kind of an infinite universe it is. Also, if you have to wait an infinitely long time for something to happen, does it actually happen?
But, putting aside my doubts, it would take an infinitely long time to organise each stage (e.g. print the tickets, or get the applications, or whatever), and that must come before the draw, so would it ever get drawn? If it did get drawn and there was an infinite number of winners, there would also be an infinite number of losers, both countably infinite, so the same order of infinity, but there's no paradox there.
It's no different in principle from the whole numbers (e.g. all the lottery players) being countably infinite, and the multiples of 1 million (e.g. one in a million are winners) also being countably infinite.
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I have often criticized the "infinite monkeys" type arguments that imply that an infinite sample size must contain all possible outcomes. I guess some of this comes down to definitions (which infinity, how does one define non-zero probability etc.)
But my counterargument is an infinite number of monkeys with an infinite amount of time punching numbers into a keypad. Even if it weren't strictly forbidden to include the digit, 9, the monkeys could generate an infinite number of numbers that did not include the 9. Just think of generating all of the possible integers, but in base 9, or base 8, or base 2. Now the probability that the monkeys would randomly not produce any numbers that included a 9 is extremely small, but there are an infinite number of ways that it could be done...
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Now the probability that the monkeys would randomly not produce any numbers that included a 9 is extremely small, but there are an infinite number of ways that it could be done...
Yes; in an infinite universe it might be infinitely unlikely (i.e. an infinitely small chance that a 9 doesn't occur), but still a possibility [;)]
I have a feeling I've heard a more robust argument, but I can't remember how it went [:(]
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If the lottery pays out at a finite rate or in a finite period though, these paradoxes can be eliminated. A million dollar lottery that draws every week would have a finite pool of tickets and ticket holders as well as finite buy-in and payout for each cycle, so even though the process repeats infinitely, the infinity isn't involved in any one game.
Agreed; but that is quite a different concept, and involves the unbounded, rather than the infinite. The finite lottery that continues indefinitely is an example of the “infinite” sequence, that may be unbounded, but could never be established to be infinite.
I have often criticized the "infinite monkeys" type arguments that imply that an infinite sample size must contain all possible outcomes. I guess some of this comes down to definitions (which infinity, how does one define non-zero probability etc.)
A boundless sample might exist that did not contain all possible outcomes, but not an infinite sample.
The main problem with the infinite number of monkeys, with an infinite amount of time is that, although it may be an interesting concept about which to speculate, in reality, like Hilbert’s Hotel, it is meaningless.
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I doubt that an infinite lottery is possible whether the universe is infinite or not (except in the sense that chiralSPO suggests).
I share your doubts, but possibly not for the same reason.
I'm also not sure whether everything possible must necessarily happen in an infinite universe - I think it depends on the event and what kind of an infinite universe it is. Also, if you have to wait an infinitely long time for something to happen, does it actually happen?
Nothing would happen in an infinite universe. You could not wait an infinitely long time for something to happen, because infinity is not a length of time.
But, putting aside my doubts, it would take an infinitely long time to organise each stage (e.g. print the tickets, or get the applications, or whatever), and that must come before the draw, so would it ever get drawn?
No, it would never get drawn; in an infinite universe in would always be in a state of having been drawn, if the concept of being drawn has any meaning in infinity.
It's no different in principle from the whole numbers (e.g. all the lottery players) being countably infinite, and the multiples of 1 million (e.g. one in a million are winners) also being countably infinite.
Countable and uncountable infinities are mathematical concepts, but that’s as far as it goes. A countable infinity is a set with the same cardinality as some subset of the set of natural numbers. All that means is that whatever the objects in the “infinite” set may be; they can be placed in one-to-one relationship with the set of natural numbers. Valuable as that concept may be in mathematics, beyond that, it is meaningless.
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It would be easy enough to design a lottery with very rare payouts.
Just have more overall numbers. So, rather than 1-50, choose 1-100, 1-1000, or 1-1,000,000
Then just increase the number of required matches. Say, with the 1-1000 numbers, require players to match 50 distinct numbers.
Even if there are a lot of players, it could take a good long time before a payout. Think of the racket of the person putting the ticket revenue in the bank and investing on the unpaid proceeds.
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Even if there are a lot of players, it could take a good long time before a payout.
I suspect there would still people who would buy thickets on the principle that the payout could just as well be on the first draw.
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Nothing would happen in an infinite universe.
I don't see the justification for that, but if so, it means your infinite lottery couldn't happen.
You could not wait an infinitely long time for something to happen, because infinity is not a length of time.
... in an infinite universe in would always be in a state of having been drawn, if the concept of being drawn has any meaning in infinity.
Presumably not, if nothing happens.
Countable and uncountable infinities are mathematical concepts, but that’s as far as it goes. A countable infinity is a set with the same cardinality as some subset of the set of natural numbers. All that means is that whatever the objects in the “infinite” set may be; they can be placed in one-to-one relationship with the set of natural numbers. Valuable as that concept may be in mathematics, beyond that, it is meaningless.
So your infinite lottery post was proposing meaningless nonsense?
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So your infinite lottery post was proposing meaningless nonsense?
EUREKA!!! Dlorde, step forward and take a bow.
The original intention of the infinite lottery scenario was to demonstrate the absurdity of trying to apply the concepts of mathematical infinities to actual infinity, whatever that may be.
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Nothing would happen in an infinite universe.
I don't see the justification for that, but if so, it means your infinite lottery couldn't happen.
The justification for saying that nothing would happen in infinity is perhaps a bit lengthy to post in a thread into which infinity was introduced by way of thread drift. If I can find some time, I might try to put together some sort of justification, and set it up for people to shoot at.
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The original intention of the infinite lottery scenario was to demonstrate the absurdity of trying to apply the concepts of mathematical infinities to actual infinity, whatever that may be.
OK, well I'm sure we can all think up applications of mathematical concepts that would be absurd in the real world, but sometimes mathematical models of the real world lead to apparently absurd conclusions that turn out to be true in the real world, such as the Copernican Revolution, quantum mechanics, or a Mobius strip.
Where possible, these apparent absurdities must be tested against real-world data, or for correspondence with other tested theories.
As I understand it, considering the geometry of the universe, it can have positive, negative, or zero spacetime curvature. Ours has been measured as zero (within experimental error), which gives it a flat local geometry. There are various possible overall 'global' geometries, the simplest and most obvious of which is Euclidean space, which implies infinite extent. There are also finite solutions, with more complex topologies. At present we don't know which might be the case, but we can't rule out a solution just because it is counterintuitive, or seems absurd.
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Where possible, these apparent absurdities must be tested against real-world data, or for correspondence with other tested theories.
Precisely, which is why it’s worth looking at things like Hilbert’s Hotel, Schrödinger’s Cat and even the “infinite lottery” just to see what emerges.
As I understand it, considering the geometry of the universe, it can have positive, negative, or zero spacetime curvature.
My understanding is the same. No problem.
One of the reasons I try to maintain a distinction between the Universe and the cosmos is that I have no problem with the use of the term infinite, either in mathematics or with regard to the Universe, provided there is clarity as to how it is being used. On the other hand, I have difficulty understanding how the cosmos can be anything other than infinite/eternal in the sense that it is timeless and possibly dimensionless in any sense that can be appreciated fully from our 3+1 dimensions.
Please be assured, I am not claiming “this is how the cosmos is”, or ever “I believe this is how the cosmos is”. I would be poorly qualified to make such a presumptuous statement. My objective is to gain understanding. If in so doing I try the patience of others, I apologise, but that doesn’t mean I will stop doing it. [;D]
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the jackpot winners (being a percentage of the infinite number of people taking part) would be infinite
You cannot make such predictions without more information.
One mathematical trick you can apply when dealing with infinities is to look at the limit as the number approaches infinity. You can then work out what is the probability of a win. And the answer is that "it depends on the rules of the lottery".
For example, if you imagine that the number of balls in the urn is equal to the number of players, and the winner must correctly identify 6 balls drawn from the urn, you find that the probability of someone winning decreases rapidly with the number of players.
So you expect zero winners in a single game following these rules, despite an infinite number of players.
I am sure you could invent other rules that provide no winners despite an infinite number of games. For example: if there are n players and m games, set the number of balls to n, and the winner must correctly identify m balls. A quick trial calculation suggests that this would produce no winners despite an infinite number of players and an infinite number of games (provided the number of players>number of games+2).
By these rules, the only winner would be the person running the lottery (this is not so different from ordinary, finite lotteries!).
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I think estimate of a 10% payback on average from the cost of a ticket was pessimistic I have always understood that for every pound you spent on tickets on average you got back 25 pence but you would have to buy tickets for a long time to achieve this.
From a financial point of view the best bet is not to buy tickets !
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Here's a fun game (it looks very simple at first, but I promise it's worth a good thought or two):
I will flip a fair coin until it comes up heads (H). However many flips it takes to get that heads will determine your payout:
If the first one is heads (H), you get $0.50; if I flip tails, then heads (TH), you get $1.00; TTH gets you $2; TTTH gets you $4, and so on. (payout = $0.05 × 2n, where n = total number of flips)
How much would you pay to play this game? What is the expected payout (for those who know how to calculate such a thing)? What is the probability of winning more than $50 in a single game?
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This is a fun game, I promise!
Here's a fun game (it looks very simple at first, but I promise it's worth a good thought or two):
I will flip a fair coin until it comes up heads (H). However many flips it takes to get that heads will determine your payout:
If the first one is heads (H), you get $0.50; if I flip tails, then heads (TH), you get $1.00; TTH gets you $2; TTTH gets you $4, and so on. (payout = $0.05 × 2n, where n = total number of flips)
How much would you pay to play this game? What is the expected payout (for those who know how to calculate such a thing)? What is the probability of winning more than $50 in a single game?
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Own up has any correspondent to this forum ever bought a lottery ticket if so why ?
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Own up has any correspondent to this forum ever bought a lottery ticket if so why ?
After a few weeks of losers, there is a point that the lottery frequently reaches where the expected payout is greater than the odds.
So, say there is a 1 in 10 Million chance of winning, but an estimated $15 Million payout. Then the cost/benefit becomes ambiguous.
Of course, the odds of winning that $15 Million is still mighty slim.
The true payout is likely less due to taxes & fees, and perhaps accepting an annuity. Nonetheless, it is a good starting point.
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Then just increase the number of required matches. Say, with the 1-1000 numbers, require players to match 50 distinct numbers.
Even if there are a lot of players, it could take a good long time before a payout. Think of the racket of the person putting the ticket revenue in the bank and investing on the unpaid proceeds.
There is a limit to the gullibility of the public. If they don't see any prizes being awarded, they will go elsewhere.
However, as long as they sometimes get a small prize (that pays for their ticket), or hear of a friend who won a small prize, it will keep them hooked, and continue throwing good money after bad.
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Own up has any correspondent to this forum ever bought a lottery ticket if so why ?
Guilty! Back in the days when tickets were £1, I tried a few times. I had 3x£10 wins in quite quick succession, but when I reckoned I had spent more than I had won, I ended that experiment.
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Good on you Alan
You made me laugh there :)
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" 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."
Actually I think you are catching on to something important there. Some combinations of numbers are indeed reducible, keeping their 'information' intact, while other number sequences are impossible to reduce and keep their information. In those cases you will need the whole sequence to transmit the 'information', and it's not only about numbers.
Eh sorry :)
But the question connected to something I read recently, although not about lottery's. And you questioning why actually made me think that you too somehow felt there should be different 'weights/importance' attached to different combinations
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An interesting thing about odds...
Say there are 6 numbers from 1 to 50, no repeats.
Then if one summed the numbers:
N1+N2+N3+N4+N5+N6
The minimum sum would be: 1+2+3+4+5+6 = 21
The maximum sum would be: 50+49+48+47+46+45 = 285
The probably of each sum from 21 to 285 is NOT the same.
There is only one way to get a sum of 21, and a sum of 285.
One way to get 22: 123457
Two ways to get 23: 123458 & 123467
Three ways to get 24: 123459 & 123468 & 123567
I assume the most common sum would be near the middle of the range, or about 153 which one could arrive out in a bunch of different ways.
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I play the lottery every week, but at zero cost. The odds of me finding the winning lottery ticket in a gutter, dropped by a careless gambler, are not practically too different from the odds of purchasing the winning ticket. Therefore, when I am out, I carefully scan gutters for that winning ticket. This affords me an opportunity to become rich, costs me nothing and has made me an expert on gutters.
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:)
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Just a thought about lottery combinations. Consider two combinations:
A. 1,2,3,4,5,6
B. 12,19,27,39,40,48
Then ask two questions:
Which has the better chance of being this week’s jackpot winner?
Which is more likely to appear on any punter’s ticket?
One thing of which we can be fairly sure is that the winning combination will look more like B than A.