[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]

Let [math]A_1[/math], [math]A_2[/math], and [math]A_3[/math] be events, and let [math]B_i[/math] represent

either [math]A_i[/math] or its complement [math]\tilde A_i[/math]. Then there are eight possible choices for the triple [math](B_1, B_2, B_3)[/math]. Prove that the events [math]A_1[/math], [math]A_2[/math], [math]A_3[/math] are independent if and only if

[[math]] P(B_1 \cap B_2 \cap B_3) = P(B_1)P(B_2)P(B_3)\ , [[/math]]

for all eight of the possible choices for the triple [math](B_1, B_2, B_3)[/math].