Two bulbs both surviving the hour

Question

Two identical devices each perform a task, started together and running independently. Each device's completion time is exponential with a 30-minute average (rate 2 per hour).

What is the probability that both devices complete their task within 1 hour?

Accepted Answer

"Both complete within 1 hour" concerns the later of the two, the maximum completion time. The two devices are independent and identical, so the joint chance is one device's complete-by-1-hour chance multiplied by itself. For an exponential with rate \lambda, the chance of completing by time t is 1 - e^{-\lambda t}. With rate \lambda = 2 per hour and t = 1 hour, \lambda t = 2. Each device's chance of completing by 1 hour is 1 - e^{-2} \approx 1 - 0.1353 = 0.8647. Multiply the two identical factors: P(	ext{both done by 1 hour}) = (1 - e^{-2})^2 \approx 0.8647^2 = 0.7476 \approx 0.748. Because …

Two bulbs both surviving the hour

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