Customers arrive at a coffee shop according to a Poisson process with a rate of = 10 customers per hour. Each customer orders exactly one beverage; fifty percent of the customers order a coffee, 30 percent order green tea, and 20 percent order herbal tea.

(a) Calculate the probability that between time t = 5 hours and time t = 8 hours that exactly 8 customers order green tea.

(b) Calculate the probability that between time t = 5 hours and time t = 8 hours that exactly 8 customers order green tea and that exactly 5 customers order herbal tea.

(c) At time t = 4 hours, 35 customers have visited the cafe. Given this, what is the probability that exactly 20 of the customers ordered coffee?