A point $P$ is randomly selected from the square region with vertices at $(\pm 2, \pm 2)$. What is the probability that $P$ is within one unit of the origin? Express your answer as a common fraction in terms of $\pi$.
I need to find the area of the region that is within one unit of the origin, and divide it by the total area of the square. The region within one unit of the origin is a circle with radius one and center at the origin. The area of a circle with radius $r$ is $\pi r^2$, so the area of this circle is $\pi \cdot 1^2 = \pi$. The area of the square is $4 \cdot 4 = 16$, since it has side length four. So the probability that $P$ is in the circle is $\frac{\pi}{16}$. 