Three randomly chosen persons are to be tested for the presence of gene \(g\). Three random events are introduced:

\(A=\) 'none of them has gene \(g\),'

\(B=\) 'at least one of them has gene \(g\), '

\(C=\) 'not more than one of them has gene \(g\) '.

Determine the corresponding sample space \(\Omega\) and characterize the events \(A \cap B, B \cup \bar{C}\), and \(\overline{B \cap \bar{C}}\) by elementary events.