Question

For safety reasons, 3 different alarm systems were installed in the vault containing the safety deposit boxes at a Beverly Hills bank. Each of the 3 systems detects theft with a probability of 0.86 independently of the others. The bank, obviously, is interested in the probability that when a theft occurs,at least one of the 3 systems will detect it. What is the probability that when a theft occurs, at least oneof the 3 systems will detect it?

Answer 1

Answer: 0.997256

Explanation:

Given : The probability that systems detects theft :
`p=\0.86`

Sample size : n= 3

Using Binomial probability formula :-

`P(X=x)=^nC_xp^x(1-p)^(n-x)`

The probability that when a theft occurs, at least one of the 3 systems will detect it

=
`P(x\geq1)=1-P(x=0)`

`1-^3C_0(0.86)^0(1-0.86)^3\\\\=1-(1)(1)(0.002744)\ [\becuase ^nC_0=1]\\\\=1-0.002744=0.997256`

Hence, the probability that when a theft occurs, at least oneof the 3 systems will detect it = 0.997256