I'm looking for guidance on how to solve one of my homework problems. Thank you in advance.
Q: 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?
From what I understand, Poisson processes model things like number of arrivals per unit of time. The relation between the Poisson process and Exponential processes is essentially that Exponential functions are "intervals" in the Poisson process (?). Based on that:
Say I use $N_t$ ~ Poisson(5 units / 30 min) where ($N_t$) = # of orders/ half hr.
Then for part a): I find the # of orders in 45 minutes (say X), then use conditional probability to solve for P(3 large and 2 small | X orders)?
Part b) Use exponential functions of large and small orders to find a minimum $\tau_i$ where $i$ is either a large or small order?
Part c) Conditional probability.
Again, thank you for any guidance that can be provided!