Series System Reliability

When $n$ components must all work, their reliabilities multiply, so $n$ identical parts at reliability $R$ give $R^n$. Ten components at 99% each give $0.99^{10} = 90.4\%$. Every part was excellent and the system is mediocre, and that arithmetic is the reason distributed systems are hard.

The rough approximation $1 - n(1-R)$ is worth carrying in your head. Ten parts each failing 1% of the time fail collectively about 10% of the time, and the estimate is good while $n(1-R)$ stays small. It falls apart when the product gets long: a hundred parts at 99% is not 0% reliable, it is $0.99^{100} = 36.6\%$, because the approximation ignores the cases where two things fail at once. Notice also that the system is always worse than its worst component, no exceptions, and that adding a dependency can never help.

Applied honestly this is uncomfortable, which is why it usually is not. A request path through a load balancer, three microservices, a cache and a database is a series system, and if each hop is 99.9% then the path is 99.5% and your four nines were fiction. The assumption to check before trusting the multiplication is independence. Components sharing a power supply, a rack, a network segment or a deployment pipeline fail together, and correlated failure makes the real number worse than the formula predicts rather than better.