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?