We define 2 value of Xn: Xn = 0 (for hot drink) and Xn = 1 (for cold drink), thus Xn is a Bernoulli process with parameter p. Let M denote the first time that a customer wants the same kind as the previous customer.

(a) Find the pmf of M.

(b) What is the probability that XM+1 = 1?

(c) If cold drinks take 30 seconds to prepare, and hot drinks take 60 seconds. What is the expected time taken to serve the first one hundred customers?

(d) The shop only has enough ice for z many cold beverages. What is the expected total number of customers served when the supply is exhausted?

(e) What is the probability that no hot drinks have been requested at (or before) the moment the ice is exhausted?