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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