Exercise 399

Let $x$ and $y$ be chosen independently at random from the interval $[0,1]$. What is the probability that $\sin^2(\pi x)+\sin^2(\pi y)>1$?

Solution

This is $P(1)$ in the notation of the problem, where we found $P(a)=2-a-\frac1{2a}$ for $\frac12\le a\le1$. So $P(1)=2-1-\frac12$.
$P(1)=\boxed{\dfrac12}$.