You work at Marriott and are tasked with figuring out how often a given hotel is vacant. Suppose that the number of guests checking into the hotel is Poisson distributed with mean 10. Also, suppose that the number of days a guest stays in the hotel is geometrically distributed with parameter 0.5. Thus, a guest who spent the previous night in the hotel will, independently of how long they have already spent in the hotel, check out the next day with probability 0.5. Suppose that guests are independent of each other. It is also given that the maximum capacity of the hotel is 3. Now answer the following questions:

1. Model the number of guests checked into the hotel as a Markov Chain.

2. What is the probability that the hotel is empty in 3 days?

3. Compute the expected number of times the hotel will be empty over the next three days, given that the hotel started with one guest.