Discussion
Suppose the lifetime of a system follows the exponential distribution with failure rate 0.8. Then the probability that the system will survive longer than twice its mean time to failure is closest to :
Answer: C
The probability is closest to 0.135.
With failure rate \(\lambda = 0.8\), the mean time to failure is \(1/\lambda = 1.25\), and twice that is 2.5. The survival function of the exponential gives
\[ \begin{aligned} P(T \gt 2/\lambda) &= e^{-\lambda \cdot 2/\lambda} \\ &= e^{-2} \approx 0.135 \end{aligned} \]
The rate cancels: for any exponential lifetime, the chance of surviving beyond twice the mean is \(e^{-2}\), and beyond the mean itself is \(e^{-1} \approx 0.368\).
None of 0.331, 0.261 or 0.101 equals \(e^{-2}\); the trap is to evaluate \(e^{-0.8t}\) at a time other than 2.5, for example the mean 1.25 itself, which gives \(e^{-1} \approx 0.368\).
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