A machine already running forty hours

Question

A machine's time to failure is exponentially distributed with a mean of 100 hours. It has already been running for 40 hours without a hitch.

Given the 40 hours of trouble-free running, what is the probability it lasts at least 200 more hours?

Accepted Answer

The exponential is memoryless, so a component that has not yet failed is, in effect, as good as new. The time already logged does not shorten the remaining wait: P(T s + t \mid T s) = P(T t). So the 40 hours already run are irrelevant, and we just need the chance a fresh exponential wait exceeds 200 hours. With mean 100 hours, the rate is \lambda = 	frac{1}{100} per hour, so for t = 200, P(	ext{at least 200 more}) = e^{-\frac{1}{100}\cdot 200} = e^{-2} \approx 0.135. Be careful: this is a model fact, not a law of physics. Real machines wear out, so their hazard rises with age and the expone…

A machine already running forty hours

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