Correlation between min and max of two uniforms

Draw two independent $U(0,1)$ values and call the smaller one $Y$ and the larger one $Z$. For $0\le y\le z\le 1$, compute $P(Y\ge y\mid Z\le z)$, and then find the correlation coefficient of $Y$ and $Z$.

$P(Y\ge y\mid Z\le z)=\left(\tfrac{z-y}{z}\right)^2$ and $\operatorname{corr}(Y,Z)=\tfrac12$. Given $Z\le z$ both draws are uniform on $[0,z]$, and both must exceed $y$. For the correlation, $YZ$ is the product of the two draws, so $E[YZ]=\tfrac14$, giving $\operatorname{Cov}=\tfrac14-\tfrac13\cdot\tfrac23=\tfrac1{36}$, while each variance is $\tfrac1{18}$.