A smoke-detector system uses two devices, $A$ and $B$. If smoke is present, it will be detected by device $A$ with probability
of $0.93$, by device $B$ with probability of $0.96$; and by both devices with probability of $0.91$.
What's the probability that the smoke will be detected by device $B$ given that it is not detected by device $A$?
I'm not quite sure how to determine the probability of device $B$, given that $A$ fails. I think there is a formula for conditions like that, but I can't find it. Any help is appreciated.
Best Answer
$\Pr(B\mid A^c) = \dfrac{\Pr(B\cap A^c)}{\Pr(A^c)}$ by definition of conditional probability.
Next, $\Pr(B\cap A^c) = \Pr(B)-\Pr(B\cap A)$ by total probability.
Finally, $\Pr(A^c)=1-\Pr(A)$, again by total probability.