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.

1 Answer
Alan P.
Jan 19, 2017

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

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