I am trying to solve the following problem.
The time $T$ required to repair a machine is an exponentially distributed random variable with mean 10 hours.
a) What is the probability that a repair takes at least 15 hours given that its duration exceeds 12 hours? b) What is the probability that the combined time to repair two machines is at least 20 hours?
Solution Attempt
Since mean is given to be 10 hours hence $\lambda = \dfrac {1}{10}$ and the probability distribution of the time is given as $e^{-\lambda t} = e^{-\dfrac {1}{10} t} $
a) $P(T>15 |T>12) = P(0 $ repairs in $ (12, 15]) = e^{-\dfrac {1}{10} 3}$
b) let $T_1$ be the r.v representing time to repair the first machine and $T_2$ be the r.v representing time to repair the second machine. So we seek to evaluate $P(T_1 + T_2 > 20)$ we know both of these time should be independent as the exponential distribution process to memory less but i am not sure how to proceed from here.
Any help would be much appreciated.