+
Write \(F\) for the event that there’s a fire and \(S\) for the event that there’s visible smoke. Then the information we’re given can be interpreted as:
+
+\begin{align*}
+\Pr(F) \amp = 0.01 \\
+\Pr(S) \amp = 0.1 \\
+\Pr(S\mid F) \amp = 0.9
+\end{align*}
+
+
In this case, we can use the simpler version of Bayes’ Theorem:
+
+\begin{gather*}
+\Pr(F \mid S) = \frac{\Pr(S \mid F)\Pr(F)}{\Pr(S)} = \dotsb
+\end{gather*}
+
+
Unlike our usual diagnostic testing examples, we do have access to the denominator probability here.
+
+