A machine already five hours between faults

Question

A machine suffers breakdowns as a Poisson process at rate $\lambda = 0.25$ per hour. It has now run $5$ hours since its last breakdown.

Given that, what is the probability it runs at least $8$ more hours without a breakdown?

Accepted Answer

The exponential time between breakdowns is memoryless, so the 5 hours already logged do not matter. The remaining run behaves like a fresh exponential wait, and we want it to exceed 8 hours: P(T 8) = e^{-\lambda t} = e^{-0.25 	imes 8} = e^{-2} \approx 0.135. About a 13.5\% chance. A machine with memoryless failures is not "wearing in" or "wearing out"; its past uptime carries no information about the next failure. Takeaway. For a memoryless failure process, remaining lifetime does not depend on age; the chance of surviving another t is always e^{-\lambda t}.

A machine already five hours between faults

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