 A subsequent measurement of \(q\) in this state then yields a positive result with probability \begin{equation} \textrm{Tr}(W_{P} Q) = \frac{\textrm{Tr}(QP W PQ)}{\textrm{Tr}(W P)} = \frac{\textrm{Tr}(W PQP)}{\textrm{Tr}(W P)}.  \end{equation} The operator \(PQP\) is not a projection unless \(P\) and \(Q\) commute, but is always an effect.  If we write \(\Pr(a|W)\) for \(\textrm{Tr}(Wa)\) for arbitrary effects \(a\), then the above can be rewritten, perhaps more transparently, as