Two real numbers are chosen at random between $0$ and $2.$ What is the probability that the sum of their squares is no more than $4?$ Express your answer as a common fraction in terms of $\pi$.
To find the probability, I need to compare the area of the region where the sum of the squares is no more than $4$ to the area of the whole sample space, which is a square of side $2.$ I can visualize the region where the sum of the squares is no more than $4$ as a quarter-circle of radius $2$ in the first quadrant, since $x^2 + y^2 \leq 4$ implies that $(x,y)$ is within $2$ units of the origin. The area of this quarter-circle is $\frac{1}{4}\pi \cdot 2^2 = \pi.$ The area of the whole sample space is $2 \cdot 2 = 4.$ Therefore, the probability is $\frac{\pi}{4}.$

