Question

For safety reasons, 4 different alarm systems were installed in the vault containing the safety deposit boxes at a Beverly Hills bank. Each of the 4 systems detects theft with a probability of 0.84 independently of the others. The bank, obviously, is interested in the probability that when a theft occurs,at least one of the 4 systems will detect it. What is the probability that when a theft occurs, at least oneof the 4 systems will detect it?

Answer 1

99.93%

Explanation:

Probability of Independent Events

Given the probability of success of each detector is 0.84 independently of the others, their combined success/failure probability can be computed with the product rule.

We can calculate the required probability by using the binomial distribution, but it's easier to calculate the probability of the negated event an subtract from 1.

We want to know the probability that a least one of the 4 systems detects the occurrence of theft. That probability is the sum of the probabilities that one of them, two of them, three of them or all of them succeed. The negated event is that NONE of them actually detects the theft. Being p the individual probability of success, p=0.84. Being q the probability of failure, q=0.16.

The probability that none of the systems detect the theft is

`P(0)=q^4=0.16^4=0.00065536`

Thus, the probability that at least one of the systems detect the theft is

`P(x\geq1)=1-P(0)=1-0.00065536=0.9993`

That means a 99.93%