[CliffordK]

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.

[evan_au]

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.

[Bill S]

I appologise if I have posted this in the past, but my interest in infinity demands an infinite lottery.  []

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.

[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.

[dlorde]

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.

[chiralSPO]

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...

[dlorde]

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 []

[Bill S]

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. 

[Bill S]

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.

[CliffordK]

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.

[Bill S]

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.

[dlorde]

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?

[Bill S]

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.

[Bill S]

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.

[dlorde]

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.

[Bill S]

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.   []

[evan_au]

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!).

[syhprum]

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 ! 

[chiralSPO]

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?

[chiralSPO]

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?