1. The number of people arriving at a bicycle repair shop follows a Poisson distribution with an average of 4 arrivals per hour. A given customer who arrives at the shop orders a repair with 70% probability. Let X represent the number of people arriving per hour.

a. What is the probability that six people arrive at the bike repair shop in a given hour? Why?

b. What is the probability that there will be at least 1 arrival within the next hour? Why?

c. What is the probability that two customers arrive within the next hour and just one of them orders a repair? Explain the steps used to arrive at your answers.

2. A restaurant manager classifies customers as regular, occasional, or new. He finds that the fraction of all customers who are regular customers is 50%, the fraction of occasional customers is 40%, and the fraction of new customers is 10%. The manager found that beer was ordered by 80% of the regular customers, 40% of the occasional customers, and 20% of the new customers.

a. What is the probability that a randomly selected customer orders a beer? (Hint: Let A be an event. Let E1, E2,..., EK be mutually exclusive and collectively exhaustive events. Then P (A) = P (A n E1)+P (A n E2)+:::+P (A nEK)).

b. If a customer orders a beer, what is the probability that the person ordering is a regular customer? (Hint: Use Bayes' Theorem) Explain the intuition behind the steps used to arrive at your answer.