The time to failure of a piece of equipment follows an exponential distribution with a mean of 1,000 hours. Take e^(−0.5) ≈ 0.61, e^(−1) ≈ 0.37 and e^(−1.5) ≈ 0.22. Given that the equipment has run for 500 hours without failure, which of the following is closest to the probability that it runs at least another 1,000 hours without failure?

Answer: B. 0.37

The exponential distribution is memoryless: the distribution of the remaining time does not depend on how much time has already passed. The probability of running at least another 1,000 hours after 500 hours is therefore the same as the probability that new equipment runs at least 1,000 hours, e^(−1) ≈ 0.37. Calculating it as a conditional probability also gives P(T > 1,500) ÷ P(T > 500) = e^(−1.5) ÷ e^(−0.5) = e^(−1). Parts in the random-failure period, where the failure rate is constant, therefore do not become more likely to fail the longer they have been used.

• A. e^(−1.5) ≈ 0.22 is the probability that new equipment runs at least 1,500 hours, which does not use the condition that it has already run 500 hours.
• B. Correct. By memorylessness, e^(−1) ≈ 0.37.
• C. e^(−0.5) ≈ 0.61 considers only the remaining 500 hours until a total of 1,000 hours, not the "another 1,000 hours" asked.