Question

For safety reasons, four different alarm systems were installed in the vault containing the safety deposit boxes at a Beverly Hills bank. Each of the four systems detects theft with a probability of 0.99 independently of the others. The bank, obviously, is interested in the probability that when a theft occurs, at least one of the four systems will detect it. What is the probability that when a theft occurs, at least one of the four systems will detect it. This probability is equal to:(a) (.99)4(b) (.01)4(c) 1- (.99)4(d) 1- (.01)4(e) 1- (.01)*4

Answer 1

The probability that at least one of the four systems will detect it is
`1-(0.01)^4`.

Explanation:

Consider the provided information.

Each of the four systems detects theft with a probability of 0.99 independently of the others.

Systems detects theft = p = 0.99

Systems fail to detects theft = q = 1-0.99=0.01

We need to find the probability that when a theft occurs, at least one of the four systems will detect it.

Pr (at least 1 systems detect theft) = 1 - Pr (none of the systems detect theft)

Pr (at least one systems detect theft =
`1-(0.01* 0.01* 0.01 * 0.01)`

Pr (at least one systems detect theft =
`1-(0.01)^4`

Hence, the probability that at least one of the four systems will detect it is
`1-(0.01)^4`.