HOTEL OVERBOOKING STRATEGY $(20$ points$)$

A hotel near the university always fills up on the evening before football games. History has shown that when the hotel is fully booked, the number of last$-$minute cancellations has the following distribution:

$\backslash$table$[[$Cancellations$,0,1,2,3,4,5,6,7,8,9,10],[$Probability$,0\mathrm{.}05,0\mathrm{.}15,0\mathrm{.}15,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}1,0\mathrm{.}1,0\mathrm{.}06,0\mathrm{.}04,0\mathrm{.}04,0\mathrm{.}01]]$

The average room rate is $$90.$ When the hotel is overbooked, policy is to find a room in a nearby hotel any to pay for the room for the customer. This usually costs the hotel approximately $$250$ since rooms booked on such late notice are expensive.

a$)$ How many rooms should the hotel overbook?

b$)$ What is the Expected profit of your optimal strategy in a$).$