Parallel System Reliability (Redundancy)

Redundancy is easier to reason about backwards. For the system to fail, every unit must fail, and if failures are independent those probabilities multiply. Three units that are each 90% reliable all fail together only $0.1^3 = 0.001$ of the time, so the system is 99.9% reliable. Three mediocre boxes became three nines.

The pattern is that each additional unit multiplies the failure probability by $(1-R)$, so with 90% units every extra box buys one more nine. That is an extraordinary return, and it is why redundancy beats reliability as an engineering strategy almost every time: two cheap disks in a mirror outperform one expensive disk, and it is not close. The economics only turn when the units stop being cheap or the failures stop being independent.

And independence is where this formula gets people killed, professionally speaking. Two power supplies on the same circuit, two servers in the same rack, two availability zones with a shared control plane, two disks from the same manufacturing batch failing in the same month. The formula will cheerfully report five nines for an arrangement whose real failure mode is a single flood, fire or expired certificate. It also silently assumes the failover works, which is a claim about a mechanism nobody exercised, and an untested standby is a component with an unknown reliability rather than a reliable one. Test the failover, or the arithmetic above is fiction.