For safety reasons, three different alarm systems were installed on the property of a famous movie star. Each of the three systems detects when a trespass occurs with a probability of 0.98 independently of the others. The movie star, obviously, is interested in the probability that when a trespass occurs, at least one of the three systems will detect it. What is the probability that when a trespass occurs, at least one of the systems will detect it?

Answer: 0.992

Step-by-step explanation:

Given : The three systems detects when a trespass occurs with a probability of 0.98 independently of the others.

Then, the probability that system failed to detect a trespass = 1 - 0.98 = 0.02

Since each system works independently,

Then, the probability that none of 3 systems will detect = 0.2 x 0.2 x 0.2 = 0.008

Then the probability that when a trespass occurs, at least one of the systems will detect it

= 1 - P (none of three systems detect)

= 1 - 0.008=0.992

Hence, the probability that when a trespass occurs, at least one of the systems will detect it = 0.992