Probability of average distance from origin of unit circle less than half

Question

Two independent points are uniformly distributed within a unit circle. What is the probability that the average of the distances from the points to the origin is less than half?

Answer 1

As the angle doesn't matter, you are picking two numbers (the radii) that are distributed between $0$ and $1$ with probability density function $P(x)=2x$ and asking whether the sum is greater than $1$. You can turn this into picking a point in the unit square $0 \le x,y \le 1$ with density $4xy$ Now integrate that over the half of the square where $x+y \ge 1$