For safety reasons, three different alarm systems were installed on the property

Question

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? (0.98)3 (0.02)3 1 (0.98)3 1 (0.02)3 3 * (0.98)1(0.02)2

Explanation / Answer

A better approach to do this problem is to first find the probability that none of the systems will detect a trespass and then subtract it from 1 to get the probability that when a trespass occurs, at least one of the systems will detect it.

P(None of the system will detect the trespass)

= 0.023

= 0.000008

Hence,

P(At least one of the systems will detect the trespass)

= 1 - P(None of the systems will detect it)

= 1 - 0.000008

= 0.999992

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