I'm at the casino, standing next to a roulette table with a \$10 minimum bet. I want to stay here as long as possible, so I'm going to repeatedly make the minimum bet until I run out of money.
I'm playing European roulette, and I'm putting my money on 28 every time. This means that with every spin, I have a 1 in 37 chance of winning \$350, and a 36 in 37 chance of losing \$10.
I only have \$20 in my pocket, so I'm almost certainly not going to be here for very long! (This is distinctly not awesome.) But, on the other hand, there is a small chance that I'll get 35 extra spins, so that's got to count for a little bit.
So, how long, on average, is my money going to last me? Three spins? Four?