Probability Fundamentals & Geometric Probability · Challenge problem

Two points $P$ and $Q$ are chosen independently and uniformly at random inside the square $[0,1]\times[0,1]$. What is the probability that the midpoint of segment $PQ$ lies inside the smaller square $\left[\tfrac14,\tfrac34\right]\times\left[\tfrac14,\tfrac34\right]$? Express your answer as a common fraction.