Hotel manager Basil Fawlty and his resourceful assistant Manuel run a 200 room hotel in a quaint little town called Tourkey on the eastern sea coast.In the absence of a respectable inn in the nearby vicinity, Mr. Fawlty enjoys ample demand at his low fare of $150 per night. Manuel notices that some customers walk into the inn requesting a room for that very evening and they are willing to pay a high fare of $500 per night.Manuel estimates that the demand for high fare on any given day follows a Normal distribution with mean 60 and standard deviation of 50.

Manuel suggests that, for each day, some rooms be reserved for the more profitable high-fare customers. To maximize expected profits, how many rooms should be 'protected' (or reserved) for high-fare customers?

Basil points out to Manuel that for every occupied room, they incur a cost of $70 in labor expenses, wear and tear, additional utilities used, etc.(This cost does not depend on whether the room is being used by high or low fare customers and is incurred only when the room is occupied.) What should be the new protection level for high fare customers to maximize profits with this information factored in?

Basil decides to protect 115 rooms for the high-fare customers. What is the probability that all 115 rooms will be occupied?