6. If we pick (uniformly) a random square of area 1 with sides parallel to the $x-$ and $y$-axes that lies entirely within the 5 -by- 5 square bounded by the lines $x=0, x=5, y=$ $0, y=5$ (the corners of the square need not have integer coordinates), what is the probability that the point $(x, y)=(4.5,0.5)$ lies within the square of area 1 ?
Solution: The upper-left corner of the unit square is picked uniformly from the square $0 \leq x \leq 4 ; 1 \leq y \leq 5$, and for it to contain the desired point it must lie in the square $3.5 \leq x \leq 4 ; 1 \leq y \leq 1.5$. The answer is the ratio of the squares' areas, $\frac{1}{4} / 16=\frac{1}{64}$.