I have been given the following question, and am unsure what type of RV to use in order to best model the problem. My initial thoughts were to use Poisson, with $\lambda = 5$. Then, the answer to a) is very easy to find. I am then confused, however, about how to solve b) & c) using Poisson.
In a pizza restaurant, customers on average select five toppings for their pizzas. However, obviously some customers prefer to have fewer or more toppings.
- a) What is the probability that the restaurant will prepare a pizza with more than 5 toppings?
- b) What is the probability that 2 of the next 3 pizzas ordered have more than five toppings?
- c) What is the probability that the fourth pizza the restaurant makes will be the first with more than 5 toppings?
You have chosen the right variable - if the number of toppings on a pizza is $X$, then $X \sim Poisson(5)$ is a fair model given the information you have.
(a) is asking for $Pr(X \geq 5)$, which requires the cumulative distribution of the Poisson. Depending on what formulas you have available to you, it may be easier for you to compute the complementary probability $Pr(X \leq 4)$ and note that the two probabilities must sum to 1.
(b) requires a bit of a two-stage thinking process. Once you solve (a), you have the probability that a single pizza will have at least 5 toppings. If that probability is $p$, then what kind of distribution could you use to model the random variable "number of pizzas out of the next 3 that have at least 5 toppings"? (Hint: What distribution do you use if you've got a fixed number of trials, and each trial has the same, independent probability of success?)
(c) is in the same ballpark. Do you know a distribution you'd use for "in a potentially endless series of trials, each with the same independent probability of success, the event that there are $n$ failures before the first success"? If you don't have one of those, can you work out the probability that the first three pizzas all fail to have 5 or more toppings, and that the 4th one succeeds?