An easy riddle from The Riddler that did not even require coding! Given that a tree changes colours at a random time A distributed according to a Uniform distribution (over (0,1)) and that it sheds its leave at a random time B distributed according to a Uniform distribution (over (A,1)), what is the time when a maximal number of trees show their new colour?
Which means optimising in t the probability that A<t<B. Which is equal to -(1-t)log(1-t) and maximal for t=1-e⁻¹, resulting in a (maximal) fraction of e⁻¹ of the trees holding to their new colour at that time.
Xi'an's Og 