Question

1. (25 pts) A home security system is equipped with three motion sensors that will trigger an alarm whenever any sensor detects movement around the house. Suppose that in the event of an unauthorized entry each sensor has a 99% chance of detecting movement independently of the other sensors. a) Find the probability that the alarm will be turned on in the event of an unauthorized entry. b) Find the probability that all three sensors will detect movement around the house in the event of an unauthorized entry.

Answer 1

(a)

The alarm will be turned on when at least one of the 3 sensors detecting the movement

P(Failure of all the 3 sensors detecting the movement) = 0.01³ = 0.000001

So,

P(at least one of the 3 sensors detecting the movement) = 1 - 0.000001 = 0.999999

So,

Answer is:

0.999999

(b)

P(All the 3 sensors detecting the movement) = 0.99³ = 0.970299

So,

Answer is:

0.970299