A chain that fails if any link breaks

Question

A safety system routes a signal through 5 components wired in series: the signal gets through only if every component works. Each component works with probability 0.98, and the components fail independently.

What is the probability the whole system passes the signal through?

Accepted Answer

In series, the system works only if all five work. By independence, multiply the individual success probabilities: P(	ext{system works}) = 0.98^5. Compute step by step: 0.98^2 = 0.9604, then 0.98^4 = 0.9604^2 \approx 0.92237, and finally 0.98^5 \approx 0.92237 	imes 0.98 \approx 0.9039. So the system works about 90.4% of the time, noticeably worse than any single 98% part. Takeaway. Series reliability multiplies, so small imperfections compound: five 98% parts in a row give ~90%, and ten would give ~82%. Adding stages in series can only lower overall reliability, the opposite of parallel re…

A chain that fails if any link breaks

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