Suppose you choose two numbers [math]x[/math] and [math]y[/math], independently at random from the interval [math][0,1][/math]. Given that their sum lies in the interval [math][0,1][/math], find the probability that

• [math]|x - y| \lt 1[/math].
• [math]xy \lt 1/2[/math].
• [math]\max\{x,y\} \lt 1/2[/math].
• [math]x^2 + y^2 \lt 1/4[/math].
• [math]x \gt y[/math].