Dumb probability question?

[GreyICE]

No. Some people here have said that, but it is untrue. The only thing close to a certainty is that you will break even eventually, but that doesn't mean that at any time you are guaranteed to be above that break even point.

I have seen a whole display of results on the roulette table end up black or red, (I know it is anecdotal, but still plausible, right?) which means some were winning big, and some were losing big. The losers will win some eventually, but may never win enough to get over the expected, break even point.

Infinity is big. Really, really big. Nope. Bigger. Not big enough yet...

Infinity is not finite. It does not follow the rules of finite systems.

 

[Baron Samedi]

Isn't Keynes also the one who said "In the long run, we're all dead"?

I've never heard this roulette story about Keynes. Was the punchline of it that Keynes thought that he could win, or that he at least wouldn't be wasting his money (it being a fair game and all)?

 

[Careyp74]

Modified, no one claimed that there was no table limit. This is a real casino we are talking about. What do you mean by the absolute distance? How is that different from the regular distance?

drkitten, I never used the word always, nor did I use the term probability 1, so I fail to see how I was making that argument. I was simply sticking to the scenario at hand.

OK, sure, I will agree that if there was no table limit, and I was not doing anything else for the next 10 years, and I had all the money in the world, I will eventually see myself at a point where I would be up n amount or down n amount (n=all rational numbers between 0 and infinity) But when there are table limits, even with unlimited funds, this wouldn't be the case.

Sol, if you look at it in the perspective of the return on such a game being 100%, you can lose the first 10 bets, and after that slowly the probability will get closer to the 100%, which I am calling breaking even. I was wrong when I said eventually you will break even, I should have said close to it.

 

[drkitten]

OK, sure, I will agree that if there was no table limit, and I was not doing anything else for the next 10 years, and I had all the money in the world, I will eventually see myself at a point where I would be up n amount or down n amount (n=all rational numbers between 0 and infinity) But when there are table limits, even with unlimited funds, this wouldn't be the case.

Only in the sense that if the table closes after losing more than 10,000 units, you will never be able to win more than 10,000 units at that table.

But the probability of your winning 9,999 units remains 1 -- even if you only ever bet one unit at a time.

Sol, if you look at it in the perspective of the return on such a game being 100%, you can lose the first 10 bets, and after that slowly the probability will get closer to the 100%, which I am calling breaking even. I was wrong when I said eventually you will break even, I should have said close to it.

No, you were right the first time.

It is a proven theorem of mathematics. If you simply bet on (fair) coin flips, one dollar for every flip, and you are limited neither by your own bankroll nor by the amount of time you and your opponent are willing to play the game, then the game will always end with your opponent hits his loss limit or goes bankrupt. (Think of it -- that's the only way the game can end.)

In particular, you are guaranteed to (eventually) be ahead of the game by any amount that the casino is willing to let you win. If you call breaking even "100%," then it's a proven theorem that you will eventually be above 100%.

 

[Alkatran]

Your strategy will determine the probability distribution over the possible end results, but in a fair game the expected change in money is always 0$, no matter what strategy you use. Note that you need to stop after *some* amount of time, since you have only a finite amount of money and time.

Consider this fair game: double-or-nothing, guess the outcome of a coin flip. Let's try a well-known strategy and see if we should expect some winning. Our strategy is simple: start with a bet of 1$, double your bet when you lose, stop when you win, and stop after n loses.

Your PDF is a likely gain of 1$ balanced by an unlikely loss of a huge amount of money. Specifically: {+1$ : p = 1-1/2^n, 1-2^n$ : p = 1/2^n}. Notice that 1*(1-1/2^n) + (1-2^n)/2^n = 0$, as expected.

Try to come up with a terminating strategy with a non-0 expected value if you need to convince yourself. Even "gambler's ruin" is balanced by highly unlikely wins of massive amounts of money.

edit: A lot of confusion is coming from assuming you have unlimited time or bankroll. These are *not good assumptions* for modeling casino play.

 

[69dodge]

Think of it -- that's the only way the game can end.

Sure, but one needs to show also that the game does end. Some games don't. For example, if the coin is biased against you, you can end up playing forever, losing more and more money as time goes on.

 

[69dodge]

[...] in a fair game the expected change in money is always 0$, no matter what strategy you use.
[...]
A lot of confusion is coming from assuming you have unlimited time or bankroll. These are *not good assumptions* for modeling casino play.

I agree; they certainly aren't realistic assumptions.

It can be interesting to think about their consequences anyway.

I don't think their presence could be said to make the game unfair.

Well, I guess the overall game is unfair in that one party, and not the other, chooses when to stop. But each coin flip is still fair.

 

[drkitten]

It is a proven theorem of mathematics. If you simply bet on (fair) coin flips, one dollar for every flip, and you are limited neither by your own bankroll nor by the amount of time you and your opponent are willing to play the game, then the game will always end with your opponent hits his loss limit or goes bankrupt. (Think of it -- that's the only way the game can end.)

In particular, you are guaranteed to (eventually) be ahead of the game by any amount that the casino is willing to let you win. If you call breaking even "100%," then it's a proven theorem that you will eventually be above 100%.

By the way, if anyone wants to see the proof, it's available here. See section 1.5 and 1.6 for details.