A student will go to school for his Final Exam and he will take a bus to school. The time for a bus to arrive at the station follows an Exponential Distribution with a rate of $1$ bus per $10$ minutes. It takes $10$ min for the bus from the station to the school. The student needs to be at school by 8:00 a.m. What is the latest time he must arrive at the bus station so that he is on time for his exam at least $95\%$ of time? (Specifying the time in the format 00:00 a.m)
My attempt
He must arrive at bus station by 7:50 am
So a bus arrives in $10$ min on average.
$$\begin{align}P(T\le t) &= 1 - e^{-t/10}\\1 - e^{-t/10} &= 0.95\\e^{-t/10}&= 0.05\\t &= -10 \cdot \ln(0.05) \\&= 29.957\text{ min}\\&\approx30\text{ min}\end{align}$$
so he must arrive $30$ min before 7:50 am
so he must arrive by 07:20 am
ANSWER : 07:20 am
I am not sure if my logic is sound in structuring the answer.