It’s peak fall foliage season in Riddler Nation, where the trees change color in a rather particular way. Each tree independently begins changing color at a random time between the autumnal equinox and the winter solstice. Then, at a random later time for each tree — between when that tree’s leaves began changing color and the winter solstice — the leaves of that tree will all fall off at once.
At a certain time of year, the fraction of trees with changing leaves will peak. What is this maximal fraction?
Since all trees have the identical, independent peak leaf peeping distribution, in this particular case it is acceptable to lose the whole forest for a single tree. Let's define the time axis where $t = 0$ is the autumnal equinox and $t = 1$ is the winter solstice. Then we can model the randome time when this tree's leaves begins changing color as $\tau_1 \sim U(0,1).$ The second random time when the leaves all drop from the tree, $\tau_2$, is also uniformly distributed, but after $\tau_1,$ so we have $\tau_2 \sim U(\tau_1, 1).$
Here the expected probability that this tree has leaves that have changed color at time $s \in (0,1)$ (which is the same as the proportion of trees with changing leaves) is \begin{align*}P(s) = \mathbb{E} \left[ \tau_1 \leq s \lt \tau_2 \right] &= \int_0^s \int_s^1 \, \frac{du_2}{1-u_1} \, du_1 \\ &= (1-s) \int_0^s \, \frac{du_1}{1-u_1}\\ & = -(1-s) \ln (1-s).\end{align*} In order to maximize this percentage, we note that the derivative is $\frac{d}{ds} P(s) = \ln (1-s) + 1,$ so the sole critical point on $(0,1)$ is $\hat{s} = 1 - e^{-1}$ which satisfies $\frac{d}{ds} P(\hat{s}) = 0.$ At this time, the maximal proportion of trees with changing leaves is $$\hat{p} = P(1-e^{-1}) = e^{-1}.$$
