Eli Bendersky's website Assume a casino roulette. In the US, a roulette has 38 numbers, 0 and 00, 13 black numbers and 13 red numbers. Let's assume for simplicity that a roulette has 50 red, 50 black and 1 green (0) numbers.
Now, on one hand it's a well known fact that the casino wins with the roulette. Say, for instance that I bet $100 on red. There's a 50/101 chance for me to win, and 51/101 to lose. In the long run, I lose, the casino wins.
But lets consider the following strategy:
I bet $1 on red. If I win, I've won a dollar. If I lose, I double my bet - that is, bet $2 on red. If I win, I take my $2 and again start betting with $1, if I lose I double my bet - $4.
Think about it... eventually, a red will appear (the chances are a little smaller than 1/2 on each throw). When it will appear, I've won exactly $1, because my current bet is the sum of all previous bets + 1.
2^N - 1 = Sigma{n = 0 to N - 1} 2^n)
In more familiar terms, think of a binary tree. In a full (complete) tree the number of leaves is always the number of all non-leave nodes + 1.
In short, in each series of betting $1, then $2, then $4, etc, UNTIL I WIN, I will eventually earn $1. And hey, the chance that NO read will appear is very low even after 4-5 throws.
So, why can't I rob the casino this way :-) ?