Calculate system reliability for series, parallel, and bridge configurations of independent components, and analyze Markov chain transition matrices. | Free MAS-I Lesson

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 $R_{i}$ ,

$R_{s} = i = 1 \prod n R_{i}$

Ten components each at 0.99 give $0.9 9^{10} = 0.9044$ . 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 $q_{i} = 1 - R_{i}$ ,

$R_{p} = 1 - i = 1 \prod n (1 - R_{i})$

Three components each at 0.80 give $1 - 0. 2^{3} = 0.992$ . Redundancy buys reliability cheaply.

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Common mistakes

Multiplying reliabilities in parallel. $R_{p} \neq = R_{1} R_{2}$ . 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

When a series chain has many components at high reliability, approximate $\prod R_{i} \approx 1 - \sum (1 - R_{i})$ for a quick check; the approximation is tight when individual unreliabilities are under 0.05. For two-state Markov chains with transition probabilities $a$ (from 1 to 2) and $b$ (from 2 to 1), the stationary distribution is $π = (b / (a + b), a / (a + b))$ ; memorize this since it appears every sitting.

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