Since everyone on Realpoor has an IQ of 160, easy question!

If you start with 1 trillion dollars and play a gambling game where your average positive expectation is 3 dollars every 100 games, with a standard devation of 20 dollars every 100 games.

Summary:

Bankroll = 1 Trillion
EV = +3/100 games
SD = 20

If you played this game for an infinite amount of time, would you eventually bust out?

If anyone is able to provide a proof of their answer which is not identical to what I find doing random google searches, I will paypal or neteller you 5 dollars.

I'm going to eventually post some explanation for this problem, but as a general hint I don't think the exact SD really matters as long as it's within certain bounds, I'm about 95% certain of that.

This solution seems way too easy, but:

If your average expectation is +$3, then the longer you play, the closer you will get to that +$3 figure. The SD is low enough not to bankrupt you on a bad streak, so, you'll never bust out.

Soriak wrote:

This solution seems way too easy, but: If your average expectation is +$3, then the longer you play, the closer you will get to that +$3 figure. The SD is low enough not to bankrupt you on a bad streak, so, you'll never bust out.

You're free to post any thoughts about whatever you think, but I'm not paying the money without proof. I'm not accepting an axiom as an answer since I believe it is not self evident.

lol, nobody wants your 5 dollars. Just post the answer.

42 - thank you

Silvermouse wrote:

lol, nobody wants your 5 dollars. Just post the answer.

I don't know the answer yet, I have a very strong suspicion but I don't want to have people agree with me for no reason.

The pure theory aspect of this question is if there is a possibility of something happening, is it destined to happen over an infinite amount of time? I happened to disagree with someone about this, I think the actual answer is a little suprising.

Mug you'll never break $10,000, so forget about all this trillion shit.

Mugaaz wrote:

The pure theory aspect of this question is if there is a possibility of something happening, is it destined to happen over an infinite amount of time? I happened to disagree with someone about this, I think the actual answer is a little suprising.

If there is a possibility for something to occur, then, given an infinite amount of attempts, it will happen. That also means the lottery could spew up the numbers 1, 2, 3, 4, 5 and 6 - but the odds are so small (for any number) that it'd take ten thousands of years to actually occur. But technically, your odds of winning with that combination are the same as any other.

Not sure what this has to do with your example though. You say that on "average" you make $3 profit after 100 games. The budget and the low margin of error essentially provide you with infinite attempts, removing any blockade from ending up with said average.

That is, of course, assuming a low bet per game. If the bet is high (ie the whole trillion) then the average profit doesn't really do you a whole lot of good, considering the odds for that bet would be only slightly better than tossing a coin.

If I understand your premesis right, then your "game" would be comparable to roulette with only one 0. Even if everyone only played black (or red), never changing their bets, the casino would, of course, make a profit, if everyone played for an infinite number of time.

And you're looking at your game from the side of the casino - slight odds in your favor and effectively unlimited amount of play.

Soriak wrote:

Mugaaz wrote: The pure theory aspect of this question is if there is a possibility of something happening, is it destined to happen over an infinite amount of time? I happened to disagree with someone about this, I think the actual answer is a little suprising. If there is a possibility for something to occur, then, given an infinite amount of attempts, it will happen. That also means the lottery could spew up the numbers 1, 2, 3, 4, 5 and 6 - but the odds are so small (for any number) that it'd take ten thousands of years to actually occur. But technically, your odds of winning with that combination are the same as any other. Not sure what this has to do with your example though. You say that on "average" you make $3 profit after 100 games. The budget and the low margin of error essentially provide you with infinite attempts, removing any blockade from ending up with said average. That is, of course, assuming a low bet per game. If the bet is high (ie the whole trillion) then the average profit doesn't really do you a whole lot of good, considering the odds for that bet would be only slightly better than tossing a coin. If I understand your premesis right, then your "game" would be comparable to roulette with only one 0. Even if everyone only played black (or red), never changing their bets, the casino would, of course, make a profit, if everyone played for an infinite number of time. And you're looking at your game from the side of the casino - slight odds in your favor and effectively unlimited amount of play.

Are you familiar with the term non sequitur? Hint: Paragraph 1 into 2.

The game example I gave was poker, and the SD is more like 17, but both of these are totally irelevant, it's just an example I'm familiar with.

The last hint I'm going to give before I spill what I think about this is that an idiot will never get the answer (wihout proof) to this question, unless purely by chance, and that chance is much less than 50%, and more like less than 1%. Smart people should see where I'm heading with this, Occulis will probably know what my suspected answer is right away.

http://www.stanford.edu/~wfsharpe/mia/rr/mia_rr1.htm

Heh don't answer it yet, I haven't had a chance to think about it.

First though, why did you quote expectation value and standard deviation over 100 games rather than one game? That throws what could be an irrelevent complication into the problem, of whether you could have pathological reward structures where there were huge disparities over single game rewards, i.e. almost identical probabilities of losing or winning a trillion dollars per game. The SD over 100 games might still be very low, but the SD per game could be very high. I'd have to think about that for awhile before I could move on to the main problem.

If you changed it to an expectation value of $.03 with SD of $.20 every game, would you still consider it to be the same problem? Or do you consider the 100 game stipulation an essential component?

sinrakin wrote:

Heh don't answer it yet, I haven't had a chance to think about it. First though, why did you quote expectation value and standard deviation over 100 games rather than one game? That throws what could be an irrelevent complication into the problem, of whether you could have pathological reward structures where there were huge disparities over single game rewards, i.e. almost identical probabilities of losing or winning a trillion dollars per game. The SD over 100 games might still be very low, but the SD per game could be very high. I'd have to think about that for awhile before I could move on to the main problem. If you changed it to an expectation value of $.03 with SD of $.20 every game, would you still consider it to be the same problem? Or do you consider the 100 game stipulation an essential component?

You can change it to those values if you like, I have no problem with that. All of the numbers were chosen at random.

Not an answer yet, but just to try to put it in numerical rather than intuitive concepts:

What we have is an infinite series of probabilities. That series could have two results: it could converge to 1, which would mean you definitely bust at some point, or it could converge to a nonzero number, like .2 or something, which would mean if you could play infinitely long, you'd have a 2/10 chance of going bust. The probability is clearly not zero, because at any point, if you take the amount of money you have at the moment, there would be a tiny probability that you could lose every bet from then on until you ran out of money. For instance, when you start the game, there's a miniscule probability that you could lose every single bet. The probability of going bust can never be less than that, so it can't be zero.

Naively, you could say that if at any point there's a fractional probability of going bust, and there are an infinite number of times you could make that check, then it must be guaranteed that you'll eventually lose. However, since your balance is increasing over time because of the positive expectation value, it's possible that that infinite series will in fact converge to a value less than one because the individual probabilites will get smaller fast enough (the longer you play, the more losses in a row you need to bust you).

Maybe that was obvious, just trying to point out that you can't simply say "if the probability is non-zero and you do it forever you're guaranteed to lose". I still suspect the answer is you'll lose because your expected bankroll appears to be going up "slowly" (i.e. linearly), but you have to compute it to be sure.

You always lose in gambling, but it's nice to fantasize that somehow you won't! There you go sir, an IQ-unrelated answer!

sinrakin wrote:

Not an answer yet, but just to try to put it in numerical rather than intuitive concepts: What we have is an infinite series of probabilities. That series could have two results: it could converge to 1, which would mean you definitely bust at some point, or it could converge to a nonzero number, like .2 or something, which would mean if you could play infinitely long, you'd have a 2/10 chance of going bust. The probability is clearly not zero, because at any point, if you take the amount of money you have at the moment, there would be a tiny probability that you could lose every bet from then on until you ran out of money. For instance, when you start the game, there's a miniscule probability that you could lose every single bet. The probability of going bust can never be less than that, so it can't be zero. Naively, you could say that if at any point there's a fractional probability of going bust, and there are an infinite number of times you could make that check, then it must be guaranteed that you'll eventually lose. However, since your balance is increasing over time because of the positive expectation value, it's possible that that infinite series will in fact converge to a value less than one because the individual probabilites will get smaller fast enough (the longer you play, the more losses in a row you need to bust you). Maybe that was obvious, just trying to point out that you can't simply say "if the probability is non-zero and you do it forever you're guaranteed to lose". I still suspect the answer is you'll lose because your expected bankroll appears to be going up "slowly" (i.e. linearly), but you have to compute it to be sure.

Pretty close to what I'm thinking. I think that even over an infinite amoutn of time there is only a finite chance of going bust.

You'd die before you went broke.

the rake will eat your bankroll up eventually

I like (1+(1/n))^n !