Problem $2$: Hilton on Time Square, New York City, has $444$ guest rooms. Hilton can sell all its hotel rooms at a discounted price of $$260$ per night to customers with a two$-$week advanced purchase. Hilton can also sell its hotel rooms at its full price of $$380$ per night to customers without advanced purchase. For any unsold hotel room, Hilton has a $75\%$ chance of selling it to some bidder from priceline.com at $$120$ per night on the day before. The number of customers who buy at full price without advanced purchase can be $100,200,300,$ or $400,$ with equal probability. Answer the following questions.

$1.$ What is the overage cost of a full$-$price room?

$2.$ What is the underage cost of a full$-$price room?

$3.$ How many rooms should Hilton reserve for the customers without advanced purchase?

$4.$ What is the probability all full$-$price customers will get a room?

$5.$ What is the expected fraction of full$-$price customers who get a room?

$6.$ What are the number of expected sales of full$-$fare customers?

$7.$ What is the number of expected sales of discounted customers?

$8.$ What is the number of expected sales from priceline customers?

$9.$ What is Hilton$$s expected revenue?