Theory

Theory 1

Consider some process schematically with components in series and components in parallel:

• Each component has a probability of success or failure.
• Event $W_i$ indicates ‘success’ of that component (same name).
• Then $P[W_i]$ is the probability of $W_i$ succeeding.

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Success for a series of components requires success of each member.

• Series components rely on each other.
• Success of the whole is success of part 1 AND success of part 2 AND part 3, etc.

Failure for parallel components requires failure of each member.

• Parallel components represent redundancy.
• Success of the whole is success of part 1 OR success of part 2 OR part 3, etc.

For series components, stack successes:

$$P[W]=P[W_1{W}_2{W}_3]=P[W_1]\cdot P[W_2]\cdot P[W_3]$$

For parallel components, stack failures:

$$\begin{array}{c}P[W^c]=P[W_1^c{W}_2^c{W}_3^c]\\ \\ \gg \gg (1-P[W_1])(1-P[W_2])(1-P[W_3])\end{array}$$

E.g. if $P[W_i]=p$ for all components $i$, then:

• Series components: $P[W]=p^3$
• Parallel components: $P[W]=1-{(1-p)}^3$

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To analyze a complex diagram of series and parallel components, bundle each:

• pure series set as a single compound component with its own success probability (the product)
• pure parallel set as a single compound component with its own success probability (using the failure formula)

This is like the analysis of resistors and inductors.