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Mathematically, we say that a symmetric random walk on a one-dimensional integer lattice will eventually reach any point with probability one. If you spend enough time playing a game where you randomly gain or lose money in such a way that each turn's expected profit exactly equals its expected loss (that is to say, the net expected profit per turn is 0), you will eventually reach any amount of money you want---$1_000, $1_000_000, $1x10^42, anything.

``Great,'' you say, ``I'm going to Vegas!'' Hold up. You see, I said any amount. That also includes $-6_000_000_000 (i.e., way in the hole). The reason this doesn't work in real life is that you have to quit when you run out of money (or, if you're foolish, run out of credit). If you keep playing, even if the game is perfectly fair (i.e. not rigged in the house's favour), you will eventually hit $0 (or $-(your credit limit)), and you have to stop. This keeps you from reaching arbitrarily high profit levels. It's also the reason casinos don't have perfectly fair games---they're in it to profit, and don't want to go bankrupt. Theoretically speaking, if your credit limit were infinite, you could reach arbitrarily hyperinflatory profit levels, since you never have to stop playing.

So, in real life, you've got to know when to walk away and know when to run. But it's even worse than that. Remember when I said casino games are biased towards the house? That makes the random walk asymmetric, meaning the theorem no longer applies. You can reach arbitrarily large losses, but there is no guarantee you will ever break even, no matter how long you play.

And now for something completely different. We were talking above about infinite symmetric random walks on a one-dimensional integer lattice. It happens to work with two-dimensional lattices, too. Things change in three or more dimensions. In three dimensions, the probability that an infinite symmetric walk will return to the point it began at is actually around 1/3. As the number of dimensions increases, the probability decreases (~0.20 for four dimensions, ~0.136 for five, ~0.105 for six, etc.). Sadly, there seems to be a dearth of practical applications of higher-dimensional random walks to Vegas gambling.

To explain for neil: Take a coin. Give yourself five 'points', and then flip the coin. If the coin lands as heads, give yourself a point. If it lands tails, take one away. The goal of the game is to stop while you have the most points possible, and the game is over when you hit zero.

This is how the gambler's dillema works: It is possible to win (by quitting while you're ahead) but because you lose instantly when you run out, there is a greater chance of having to stop at that point.

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