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A system consists of four components ${1,2,3,4}$. There are two routes ${1,2,4}$ and ${1,3,4}$ to complete the system. If all four components have equal reliability of $0.9$, what is the reliability of the system at a given time instant?

My answer:

I know how to calculate the reliability of the two routes, $(.9)^3$ and $(.9)^3$, but how do you combine these two to get a total reliability?

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up vote 2 down vote accepted

We assume independence. This may not be entirely realistic, but we need some assumption or information about possible interactions between the components.

The system works if and only if $1$ and $4$ work and at least one of $2$ or $3$ works.

The probability that $2$ or $3$ both fail is $(0.1)(0.1)$. So the probability at least one works is $1-(0.1)^2=0.99$.

Thus our system reliability is $(0.9)(0.99)(0.9)$.

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