Two equal-speed chores both done

Question

Two chores are started at the same time and handled independently. Each chore's completion time is exponential with a 10-minute average.

What is the probability that both chores are finished within 10 minutes?

Accepted Answer

"Both finished within 10 minutes" refers to the later of the two, the maximum. The two chores are independent, so the joint chance is the product of the individual finish-by-10-minutes chances. Here the two chores share the same rate, so the two factors are equal. For an exponential with rate \lambda, the chance of finishing by time t is 1 - e^{-\lambda t}. With a 10-minute average, \lambda = 	frac{1}{10} per minute, and for t = 10 we get \lambda t = 1. Each chore's chance of finishing by 10 minutes is 1 - e^{-1} \approx 0.6321. Because the two are independent and identical, multiply this by…

Two equal-speed chores both done

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