Gamble At/With Risk
Suppose you are gambling, starting with a small sum such as $100. You need to turn this into $200,000 by dawn, or the loan sharks will eliminate you. But, your only available game is betting Red or Black on a Roulette Wheel, which has unfavorable odds.
What is your best betting strategy?
Patrick Billingsly, renowned probabilist, suggests that one's only option is "Bold Play." At each stage, you need to bet your entire current fortune, unless a win would take you past your goal of $200,000, in which case you bet just enought to reach goal.
Billingsley showed that the Bold Play strategy maximizes the probability of reaching your goal...and living for another day or bet.
And the odd thing....to provide the mathematics necessary to justify his claims about Bold Play, Billingsley constructs a function that is both continuous and increasing...but it lacks a derivative almost everywhere...akin to the Weierstrauss function.
Once again, the amazing twists and turns of mathematics. Betting as if one's life depends on it...hinges on understanding one of those pathological real-valued functions.
Source: P. Billingsley, "The Singular Function of Bold Play." American Scientist (1983), 71:392-397.