I offer you a game where you pay $1 to have a 99/100 chance of winning $1000 and a 1/100 chance of losing your wager. The geometric mean of the possible outcomes is zero, so it would be foolish to play in such a game if you repeatedly bet your bankroll(?)
You can argue that playing an infinite number of times would guarantee that you lose your wager, but for any sensible number of plays it'd be a no-brainer. I'd even argue that after a Large number of plays, the infinitesimal chance you'll win those games times the payoff if you do win (which is very large as far as large numbers go) would make the game worthwhile.
Edited by plasmid, 23 November 2013 - 08:36 AM.