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Some mate of mine is some casino lover, and he usually says something like this to justify his hobby.

"Let's suppose we have a game, in which I gamble something, and if I win, I receive the double, and if I lose I don't receive anything. So I gamble 1 dollar. If I win, I stop gambling. If I lose, I gamble 2 dollars. If I win now, I stop gambling, but if I lose, I gamble 4 dollars, and so. Then in the end, when I win I will always have won 1 dollar".

I see a few problems with that reasoning. What if he keeps losing until he does not have more money? What does he do then?

Another way for me to think it is, let's suppose that the game is not fair, so he has a negative expected value for this game. Independently of how he gambles, he'll tend to lose, and if the only way for it to happen is just not winning even once, it will happen eventually.

So, is there any other problem with that reasoning?

Sorry, I don't know which tag to use here, I think probability may not be too appropiate.

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    $\begingroup$ You may wish to read or refer him to pages such as this one which discuss this idea of "Gambler's Ruin." If he plays with an infinite amount of money (and the house can match his bets), yes, he will have won a huge amount in the end, however if he plays with a finite amount, if he continues to play, he will eventually go bankrupt. $\endgroup$ – JMoravitz Jun 21 '15 at 3:21
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    $\begingroup$ I think this is a legitimate proof of the fact that, if you already have infinite wealth then you can win a dollar. Of course, I can imagine some more profitable things to do with your infinite wealth, and I can't imagine why you'd want more. $\endgroup$ – Andreas Blass Jun 21 '15 at 3:22
  • $\begingroup$ I think the main difficulty is that it doesn't take many losses and doubling, before the gambler runs out of money to bet, and has to quit. Thus he faces a small chance of losing a huge amount of money, which makes the overall expected loss positive. (Think of the old story of one grain of corn on the first square of a chess board, two on the second, four on the third, .... How many grains of corn on the last?) $\endgroup$ – BruceET Jun 21 '15 at 3:26
  • $\begingroup$ This is the Martingale betting system $\endgroup$ – Conrad Turner Jun 21 '15 at 5:55
  • $\begingroup$ This is also the reason that casinos have a limit on the number of times you are allowed to go "double or nothing". $\endgroup$ – DanielV Jun 21 '15 at 8:54
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This is certainly not formal, but maybe it will help you understand the system flaws a bit better.

Consider $X$ to be the waiting-time random variable representing the time until you finally win a bet and profit.

If (as mentioned in the comments) you have an infinite supply of money to bet with, then so long as $X$ is a proper random variable, i.e. $$P(X < \infty) = 1$$ you will eventually profit with a net gain of 1 unit. With an infinite supply of money, no casino limits, an unlimited amount of time to continue gambling, etc. one could theoretically win as much money as they desired with this strategy given any game where $X$ is a proper random variable, simply by adjusting their first bet (representing their unit).

Realistically however, it doesn't work out so nicely. In order for an individual to assure themselves a profit, they would need $X$ such that $$P(X<k) = 1$$ where $k$ represents the limiting number of gambles before they either

  • run out of money to gamble with
  • run out of time in which to gamble
  • ...any other constraint on how long they can gamble that may apply

You can begin to now see the flaws with this system. Finite supplies of money and time as well as other constraints render $P(X<k)<1$, and expose the gambler to the risk of losing many more units of money than the 1 that they are attempting to win with this system.

Even if an individual has enough money and time such that they can assure themselves a profit, their first bet (i.e. unit they are attempting to gain) is more than likely so small that by the expected time at which they win their returns will be minimal, either as a result of how small they were forced to make their first bet, or how long it took them to finally win.

So in short, I presume unless your friend is incredibly rich, or attempting to win some small, small amount of money, or some combination of the two, the odds are still against him.

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