How Many Backups Do You Need?

A chain fails if any one link fails; a bank of backups fails only if every one fails. With independent failures, the multiplication rule says exactly how much each extra component helps, or hurts.

How it works

Independent events: the multiplication and complement rules

In one sentence

Events $A$ and $B$ are independent when $P(A \cap B) = P(A)\,P(B)$ . For $n$ independent components that each work with probability $p$ , the number working is binomial, so $P(\text{at least } k \text{ work}) = \sum_{j=k}^{n}\binom{n}{j}p^j(1-p)^{n-j}$ ; $k = n$ gives $p^n$ , and $k = 1$ gives $1-(1-p)^n$ by the complement rule.

General formula

P(\text{all } {\htmlData{sym=1}{n}} \text{ work}) = {\htmlData{sym=0}{p}}^{\htmlData{sym=1}{n}}, \qquad P(\text{at least one works}) = 1 - (1-{\htmlData{sym=0}{p}})^{\htmlData{sym=1}{n}}, \qquad P(\text{at least } {\htmlData{sym=2}{k}} \text{ work}) = \sum_{{\htmlData{sym=3}{j}}={\htmlData{sym=2}{k}}}^{\htmlData{sym=1}{{n}}}\binom{\htmlData{sym=1}{{n}}}{\htmlData{sym=3}{{j}}}{\htmlData{sym=0}{p}}^{\htmlData{sym=3}{j}}(1-{\htmlData{sym=0}{p}})^{{\htmlData{sym=1}{n}}-{\htmlData{sym=3}{j}}}

Requiring more parts multiplies small chances of trouble into a big one; adding spares multiplies small chances of total failure into a tiny one.

What the symbols mean

$p$
Component reliability
The chance one component works, independently of the others.
$n$
Number of components
How many identical parts the system has.
$k$
Parts needed
How many of them must work for the system to work.
$j$
Parts working
A possible count of working components, from k up to n.

If failures are independent, the multiplication rule gives $P(\text{all work}) = p \cdot p \cdots p = p^n$ . A series system needs all of them, so every added part makes it less reliable.
A parallel system fails only if every part fails, which has probability $(1-p)^n$ . The complement rule turns that into $1 - (1-p)^n$ .
Many real systems need $k$ of $n$ parts working, like 2 of 3 flight computers. Counting the ways that can happen gives a binomial sum.
All of this assumes independence. A shared cause, like one power supply feeding every backup, breaks the multiplication rule and the redundancy with it.

Learn the math

The Addition and Multiplication RulesStats · Lesson 10, Probability The Binomial DistributionStats · Lesson 15, Discrete Random Variables Series, Parallel and k-out-of-n SystemsEng Stats · Lesson 11, Reliability Engineering