Suppose that the number of orders per hour at a pizza shop follows a Poisson process with rate 5 per 30 minutes. Suppose that the pizza orders are large with probability 2/3, small with probability 1/3, and the size of the pizza order is independent of the time of the call.
a) What is the probability that in 45 minutes, exactly 3 large and 2 small orders will be made?
b) What is the probability that 5 large orders will be made before 4 small orders are made?
c) Given that exactly 8 orders are made within the first hour, what is the probability that exactly 2 large and 2 small orders were made in the first half hour?
What I tried: should lambda change in terms of probability and time? Lambda will be rate 7.5*2/3 = 5 for large orders; For small orders it will be: 2.5 Am I correct?