The time-to-failure $T$ of an aircraft auxiliary power unit follows an exponential distribution with a failure rate of $\lambda = 0.002\text{ failures per flight hour}$. Given that the unit has already operated for $1{,}000\text{ hours}$ without failure, the conditional probability that it will survive for at least another $500\text{ hours}$ is closest to:

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### 1. Strategy & Theoretical Principles The exponential distribution is the only continuous probability distribution that possesses the **memoryless property**: $$P(T > s + t \mid T > s) = P(T > t)$$ This means past operating history does not affect future failure probability. ### 2. Analytical Formulation & Calculation Given: - Failure rate $\lambda = 0.002\text{ hr}^{-1}$ - Given survival $s = 1{,}000\text{ hr}$ - Additional survival required $t = 500\text{ hr}$ By the memoryless property: $$P(T > 1500 \mid T > 1000) = P(T > 500) = e^{-\lambda t}$$ $$P(T > 500) = \exp(-0.002 \times 500) = \exp(-1.0) = e^{-1} \approx 0.36788 \approx 0.368$$ ### 3. Distractor Forensics - Candidate calculating $P(T > 1500) = e^{-3.0} = 0.0498$ fails to apply the conditional probability definition. ### 4. Exam Trap & Standard Reference - **Trap Alert:** The memoryless property states that an old component with exponential failure distribution is statistically as good as new. - **Standard Reference:** Kreyszig, E., *Advanced Engineering Mathematics*, Chapter 24.

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