Question

The security system in a house has two units that set off an alarm when motion is

detected. Neither one is entirely reliable, but one or both always go off when there is

motion anywhere in the house. Suppose that for motion in a certain location, the

probability that detector A goes off and detector B does not go off is $0.25$, and the

probability that detector A does not go off is $0.35$. What is the probability that detector

B goes off?

$$\begin{gathered} a.0.1 \\ b.0.35 \\ c.0.4 \\ d.0.65 \\ e.0.75 \end{gathered}$$

Short answer

The correct option is: $e.0.75$

Step-by-step solution

Given information

Probability that detector will not go off is $0.35$.

Probability that A will go off and B will not go off is $0.25$

Explanation for correct option

The probability that detector will go off is:

$$\begin{gathered} P(DetectorAwilgooff)=1-P(DetectorAwillnotgooff) \\ =1-0.35 \\ =0.65 \end{gathered}$$

The probability that both A and B will go off is:

$$\begin{gathered} P(DetectorAandBwillgooff)=0.65-0.25 \\ =0.40 \end{gathered}$$

The probability that detector B will go off is calculated as:

$$\begin{gathered} P(Bwillgooff)=0.40+0.35 \\ =0.75 \end{gathered}$$

Thus, the required probability is $0.75$.

The correct option is (e).