A company has a security system comprising four electronic devices (A, B, C, and D) which operate independently. Each device has a probability of failure of 0.1. The four electronic devices are arranged so that the whole system operates properly if at least one of A or B functions and at least one of C or D functions. What is the probability that the whole system functions properly?

Question

anonymous · Answer

So there is a probability of .1+.1=.2 that A or B will fail. Likewise, there is a probability of .1+.1=.2 that C or D will fail. So there is a (1-.2)(1-.2)=.64 chance that the system works. Basically when you see or you add probability and when you see and you multiply probability. Someone with more knowledge in probability than I can expand upon this if you want to know the intricacies.

ybarrap · Answer

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Probability of failure is if A and B fail OR C and D fail
Probability of A and B failing is $0.1^2$.
Probability of C and D failing is $0.1^2$.

Probability that system functions is 1 - Probability of Failure = 1 - (Probability of A and B failing OR Probability of C and D failing) = $1-2\times.1^2=0.98$
$$
Therefore, the probability that the system functions properly is 0.98.