What's the probability that the smoke will be detected by device $B$ given that it is not detected by device $A$?

Question

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.

Answer 1

$\Pr(B\mid A^c) = \dfrac{\Pr(B\cap A^c)}{\Pr(A^c)}$ 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.

Answer 2

In general, given two events $A$ and $B$, the probability of $A$ given $B$ occurs is $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$. Of course, we need that $P(B)>0$ for this.