[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Choose independently two numbers [math]B[/math] and [math]C[/math] at random from the interval [math][0,1][/math] with uniform density. Note that the point [math](B,C)[/math] is then chosen at random in the unit square. Find the probability that

• [math]B + C \lt 1/2[/math].
• [math]BC \lt 1/2[/math].
• [math]|B - C| \lt 1/2[/math].
• [math]\max\{B,C\} \lt 1/2[/math].
• [math]\min\{B,C\} \lt 1/2[/math].
• [math]B \lt 1/2[/math] and [math]1 - C \lt 1/2[/math].
• conditions (c) and (f) both hold.
• [math]B^2 + C^2 \leq 1/2[/math].
• [math](B - 1/2)^2 + (C - 1/2)^2 \lt 1/4[/math].