In some Olympic sports, the athletes compete for the highest score on a trick. The judges score the trick based on how well it is completed as well as its level of difficulty. As a result, the competitors try the hardest trick they think they can successfully complete. How hard should they try?
Let's call the difficulty of the attempted trick a number 0<d<1.
The attempt is a randomly chosen from the uniform distribution 0<a<1.
If a<d the trick fails and no points are scored.
If a>d the trick succeeds and scores d*a.
What difficulty should be attempted for the highest expected score?
The probability of a trick suceeding is (1-d).
If it suceeds, a is between d and 1, so the expected value of a given that the trick suceeded is (1+d)/2.
Then the expected score is (1-d)(d)(1+d)/2 = (d-d^3)/2
The derivative with respect to d is (1-3d^2)/2
Setting this equal to 0 gives (1-3d^2) = 0.
d^2 = 1/3
d = 1/sqrt(3) = .57735 (final answer)
Expected score = sqrt(3)/9 = .19245