(related to Proposition: Probability of the Complement Event)
Because the events \(A\) and \(\overline A\) are collectively exhaustive, we have \(\Omega=A\cup \overline A\). Because they are also mutually exclusive, (i.e. \(A\cap \overline A=\emptyset\)), it follows from the definition of probability that
\[1=p(\Omega)=p(A\cup \overline A)=p(A)+p(\overline A),\] which is equivalent to \[p(\overline A)=1-p(A).\]
