If we choose $a,b,c$ randomly from $(0,1)$, is the probability $p$ that $a^2+b^2+c^2\le 1$ :
a) equal to $\pi/8$, b) smaller than $0.55$, c) larger than $0.5$, d) a rational number.
This is a weird problem a friend send to me. I don't really know how to approach this problem.
My idea was to note that picking three points $a,b,c$ from $(0,1)$ is the same as picking a point $P=(a,b,c)$ such that $a,b,c\in (0,1)$ in 3D-space. Then, $a^2+b^2+c^2$ is the length of the radius $\overline{OP}$ of one-eight of the sphere centered at the origin $O$. We want $\overline{OP}\le 1$.
Edit: is the probability equal to $$P=\frac{\frac{1}{8}\cdot \frac{4}{3}\cdot \pi\cdot 1^3}{1^3}=\frac{2}{3}\pi?? $$
Your idea is fine, but there is an error in the computation.