You own a fast food shop, which also has a drive$-$thru window. Every day when you open the shop, you will wait on average for $30$ minutes before the first customer walks in$,$ and on average $15$ minutes before the first vehicle comes to the drive$-$thru. Suppose both waiting times follow exponential distributions and they are independent. In the following questions, if you are using normal approximations, leave your final answer in terms of the standard normal CDF$.$

a$)$ What is the average waiting time for your first customer of the day $($either walk$-$in or drive$-$thru$)?$

b$)$ When the waiting time for the first customer exceeds $20$ minutes, you call it a bad day. Approximate the probability that you will have more than $60$ bad days in a year of $365$ days. Justify your use of this approximation.

c$)$ When the waiting time for the first customer is shorter than $3$ seconds $($i$.$e$.,0\mathrm{.}05$minutes$),$ it is a great day. Approximate the probability that you will have exactly $3$ great days in a year.