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?.

Answer 1

The probability that detector B goes off is 0.75

The probability of an event is known to be the likelihood or chance for an event to occur.

From the given information let consider:

• A = the time when detector A goes off, and
• B = the time when detector B goes off

Since one or both of them always goes off, then:

• P(A or B) = 1

∴

Their complements will be zero, i.e.

• P(A' and B') = 0

Similarly, we are given that:

• P(A and B') = 0.25

Then;

• P(A') = 0.35

∴

The set probability for A will be:

P(A) = 1 - P(A')

P(A) = 1 - 0.35

P(A) = 0.65

Finally, the probability that detector B goes off can be computed as:

P(B) = P(A) P(B|A) +P(A') P(B|A')

P(B) = P(A) (1 - P(B|A')) + P(A') (1 - P(B'|A')

$\mathbf{P(B) = P(A) (1-\dfrac{P(A \ and \ B')}{P(A)}) +P(A') (1 - \dfrac{P(A' \ and B')}{P(A')})}$

$\mathbf{P(B) = 0.65 (1-\dfrac{0.25}{0.65}) +0.35 (1 - \dfrac{0}{0.35})}$

P(B) = 0.75

Learn more about the probability of an event here:

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