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Author Topic: Impossible lottery combinations  (Read 20419 times)

Offline Musicforawhile

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Re: Impossible lottery combinations
« Reply #25 on: 09/11/2014 21:27:50 »

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.
 

Offline dlorde

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Re: Impossible lottery combinations
« Reply #26 on: 09/11/2014 23:49:46 »
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.
 

Offline jeffreyH

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Re: Impossible lottery combinations
« Reply #27 on: 10/11/2014 00:43:34 »
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.
 

Offline percepts

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Re: Impossible lottery combinations
« Reply #28 on: 10/11/2014 06:17:00 »
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.
 

Offline evan_au

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Re: Impossible lottery combinations
« Reply #29 on: 10/11/2014 10:50:15 »
Quote
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.
« Last Edit: 10/11/2014 10:52:29 by evan_au »
 

Offline percepts

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Re: Impossible lottery combinations
« Reply #30 on: 11/11/2014 01:10:51 »
Quote from: evan_au
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.
 

Offline alancalverd

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Re: Impossible lottery combinations
« Reply #31 on: 11/11/2014 11:28:25 »
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!   
 

Offline syhprum

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Re: Impossible lottery combinations
« Reply #32 on: 11/11/2014 15:52:39 »
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.
 

Offline burning

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Re: Impossible lottery combinations
« Reply #33 on: 11/11/2014 19:28:30 »

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?
 

Offline Musicforawhile

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Re: Impossible lottery combinations
« Reply #34 on: 11/11/2014 19:42:55 »

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."
 

Offline dlorde

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Re: Impossible lottery combinations
« Reply #35 on: 12/11/2014 00:27:57 »
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.
 

Offline CliffordK

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Re: Impossible lottery combinations
« Reply #36 on: 13/11/2014 19:26:52 »
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.
« Last Edit: 13/11/2014 19:30:10 by CliffordK »
 

Offline Bill S

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Re: Impossible lottery combinations
« Reply #37 on: 13/11/2014 19:38:37 »
Quote
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?
 

Offline alancalverd

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Re: Impossible lottery combinations
« Reply #38 on: 13/11/2014 22:59:15 »
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.
« Last Edit: 13/11/2014 23:05:10 by alancalverd »
 

Offline dlorde

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Re: Impossible lottery combinations
« Reply #39 on: 13/11/2014 23:59:03 »
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.
 

Offline CliffordK

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Re: Impossible lottery combinations
« Reply #40 on: 14/11/2014 07:13:13 »
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.
 

Offline evan_au

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Re: Impossible lottery combinations
« Reply #41 on: 14/11/2014 10:12:21 »
Quote from: CliffordK
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.
« Last Edit: 14/11/2014 10:46:18 by evan_au »
 

Offline Bill S

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Re: Impossible lottery combinations
« Reply #42 on: 14/11/2014 18:27:09 »
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.
 

Offline chiralSPO

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Re: Impossible lottery combinations
« Reply #43 on: 14/11/2014 18:40:08 »
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.
 

Offline dlorde

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Re: Impossible lottery combinations
« Reply #44 on: 14/11/2014 19:44:41 »
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.

 

Offline chiralSPO

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Re: Impossible lottery combinations
« Reply #45 on: 14/11/2014 21:22:21 »
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...
 

Offline dlorde

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Re: Impossible lottery combinations
« Reply #46 on: 14/11/2014 21:31:34 »
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 :(
« Last Edit: 14/11/2014 21:33:40 by dlorde »
 

Offline Bill S

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Re: Impossible lottery combinations
« Reply #47 on: 15/11/2014 19:52:11 »
Quote from: chiralSPO
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.


Quote
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. 
 

Offline Bill S

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Re: Impossible lottery combinations
« Reply #48 on: 15/11/2014 19:54:10 »
Quote from: dlorde
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. 

Quote
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. 

Quote
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.

Quote
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.
 

Offline CliffordK

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Re: Impossible lottery combinations
« Reply #49 on: 15/11/2014 21:35:52 »
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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Re: Impossible lottery combinations
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