Question

A home has installed 4 security systems to detect attempted break-ins. What is the probability that at least one of the 4 security systems will detect it?

They operate independently, and each has a .40 probability of detecting a break-in.

Answer 1

`color(green)(87.04%)`

Explanation:

Probability of at least one system detecting a break-in
`color(white)("XXX")= 1 -` (probability of none detecting a break-in)

If each system has a probability of `0.40` of detecting a break-in
then each system has a probability of `1-0.40 =0.60` of not detecting a break-in.

The first system will fail to detect a break-in `0.60` of the time.
Of that `0.60` of the time that the first system fails,
`color(white)("XXX")`the second system will fail `0.60` of the time.
Therefore both of the first 2 systems will fail `0.60xx0.60=0.36` of the time.

Similarly the first 3 systems will fail `0.36xx0.60=0.216` of the time;

and all 4 systems will fail `0.216xx0.60=0.1296` of the time.

Therefore at least one of the systems will succeed `1-0.1296=0.8704=87.04%` of the time