Problem - 2511
Randomly select $3$ real numbers $x$, $y$, and $z$ between 0 and 1. What is the probability that $x^2 + y^2 + z^2 > 1$?
This is a typical geometric probability problem. The possible region is the unit cube and the qualified region is the space outside the sphere. Hence the result is $$1-\frac{1}{8}\cdot\Big(\frac{4}{3}\pi\Big)=\boxed{1-\frac{\pi}{6}}$$
