Calculate system reliability for series, parallel, and bridge configurations of independent components, and analyze Markov chain transition matrices.

A reinsurer rates a satellite with five redundant subsystems and needs the probability the whole craft survives a year. Same math, smaller stakes: a Markov model for policy lapse asks what fraction of a cohort stays in force after ten renewals. Reliability and Markov chains share one engine, the multiplication of independent probabilities arranged into a structure.

Series systems. Components are in series when every one must work for the system to work. With independent components and individual reliabilities $R_i$,

$$R_s = \prod_{i=1}^{n} R_i$$

Ten components each at 0.99 give $0.99^{10} = 0.9044$. Adding cheap parts to a series chain is expensive in reliability.

KEY: In series, multiply reliabilities. In parallel, multiply unreliabilities. The two operations are duals.

Parallel systems. Components are in parallel when the system works if at least one component works. With independent unreliabilities $q_i = 1 - R_i$,

$$R_p = 1 - \prod_{i=1}^{n}(1 - R_i)$$

Three components each at 0.80 give $1 - 0.2^3 = 0.992$. Redundancy buys reliability cheaply.

Common mistakes

• Multiplying reliabilities in parallel. $R_p \ne R_1 R_2$. The correct rule multiplies unreliabilities: $R_p = 1 - q_1 q_2$. Two components at 0.80 give 0.96 in parallel, not 0.64.
• Enumerating bridge paths without inclusion-exclusion. Adding the four path probabilities double-counts shared components. Either condition on $C$ or apply inclusion-exclusion rigorously.
• Transposing the transition matrix. Some texts use column-stochastic conventions. On MAS-I, rows sum to 1 and distributions multiply $P$ from the left.

Bottom line

• Series: $R_s = \prod_i R_i$. Multiplies survival probabilities, so the weakest link dominates and reliability falls fast as components are added.
• Parallel: $R_p = 1 - \prod_i (1 - R_i)$. Redundancy multiplies failure probabilities, not survival probabilities.
• Bridge: 5-component diamond with a crossover. Use conditioning on the bridge element $C$ for a clean two-line calculation.
• Markov transition matrix $P$: nonnegative entries, rows sum to 1, entry $P_{ij}$ is the one-step probability of moving from state $i$ to state $j$; $n$-step probabilities live in $P^n$.

Exam shortcut

For a quick check on a series chain, approximate $\prod R_i \approx 1 - \sum (1 - R_i)$ when the sum of component unreliabilities $\sum_i (1 - R_i)$ is much smaller than 1, so products of unreliabilities are negligible. For $n$ identical components with unreliability $q$, use $nq \ll 1$; small individual unreliabilities alone are insufficient.

The full lesson (about 2,926 words, 20 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• A8

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