HMMT 2024 U35

Barry picks infinitely many points inside a unit circle, each independently and uniformly at random, $P_{1}, P_{2}, \ldots$ Compute the expected value of $N$, where $N$ is the smallest integer such that $P_{N+1}$ is inside the convex hull formed by the points $P_{1}, P_{2}, \ldots, P_{N}$.

Submit a positive real number $E$. If the correct answer is $A$, you will receive $\lfloor 100 \cdot \max (0.2099-|E-A|, 0)\rfloor$ points.