Series & Parallel Reliability, explained.
Series and parallel reliability distinguish a system requiring every component to work from a system requiring at least one functioning path.
Why it happens
Under independence, multiplying success probabilities gives the all-working series probability. Multiplying failure probabilities gives the all-failed parallel probability, whose complement is system success.
A series system needs every component to work; a parallel system needs at least one. Independent redundancy can improve reliability while extra required dependencies reduce it.
Read the result
Compare identical components at the same reliability and period. Adding components hurts the required chain but helps the independent fallback arrangement. Read the independence caveat before applying that comparison to real backups.
A worked example
Three components at 90% reliability
A series chain works with probability 0.9³=0.729.
A parallel arrangement fails only if all three fail: 0.1³=0.001.
Its success probability is 0.999, provided the three failures are genuinely independent and any working component is sufficient.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
For n identical independent components with success probability p, series reliability is p^n and parallel reliability is 1-(1-p)^n over the same period.
Where this idea is useful
A practical use
A service that requires every link is different from one with genuinely independent fallback paths.
A common misconception
“Three copies guarantee independent protection.”
Copies may share power, location, configuration or upstream services. A common cause can defeat all of them together.
What this explanation leaves out
- Common-cause failures violate independence. These probabilities are for one specified period, not a lifetime or repair model.
Are these probabilities lifetimes?
No. They describe working over the same specified period, with no repair or switching failures modeled.
What shared cause could disable your apparently separate backups?
Further reading
Explore the original research or the teaching reference behind this experiment.