The Gambler's Ruin

A game of blackjack is a good example of a random walk. There is some good theoretical material on this available online (see p. 487).

An example is illuminating. Assume you start with $10. Each game, you have a 49% chance of winning $1 and a 51% chance of losing a dollar. (Real blackjack odds are better.) Say you decide to play until you have $11 or go broke, whichever comes first. In this case, you only have an 11% chance of going broke and an 89% chance of winning the $11. But if you play until $15, you have a 40% chance of going broke. Play to $20 and it increases to 60%. The longer you play, the lesser your expected value. Play forever--greedy you!--and you have a 100% chance of going broke. This is known as the gambler's ruin.