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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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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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