Not long ago I was surprised to learn that when trying to maximize the expected long-term growth rate of your money, it is sometimes necessary to bet on an outcome that has negative expected value (in addition to outcomes that have positive expectation). See https://www.physicsforums.com/showthread.php?t=637064. I think I get that now, although I still don't consider it as obvious as everyone else seemed to. I'm used to problems where you are trying to maximize the total expected value on one round of betting. Surely, I thought, you would never bet any amount of your money on an outcome with negative expectation if you were trying to maximize your total expected value. Can you think of an exception to this? Can you think of a situation where there are multiple outcomes to bet on, and in order to maximize your expected value (not expected rate of return or longterm growth rate) you must bet some fraction of your money on an outcome with negative EV, along with the other positive EV bets you make?