A machine running past thirty hours

Question

A machine suffers random breakdowns, with the time between failures exponentially distributed and averaging 20 hours. You just repaired it and switched it back on.

What is the probability that it runs more than 30 hours before the next breakdown?

Accepted Answer

For an exponential time-to-failure, the chance of surviving past time t without breaking is P(T t) = e^{-\lambda t}, with \lambda the failure rate (average failures per unit time). The rate is one failure per average lifetime, so \lambda = 	frac{1}{20} per hour. For t = 30 hours, P(T 30) = e^{-\frac{1}{20}\cdot 30} = e^{-1.5} \approx 0.223. In average-lifetime units, 30 hours is 30/20 = 1.5 mean lifetimes, so the exponent is -1.5 and the survival chance is about 22%. Running well past the average lifetime is possible but increasingly unlikely. Takeaway. Survival past time t is e^{-\lambda t}…

A machine running past thirty hours

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