Probability Limited Stock Covers Every Customer in a Queue

A queue of $N$ customers is waiting in a store. Each customer will buy exactly one item: product A with probability $p$, or product B with probability $q = 1 - p$, independently of everyone else. The store has only $M$ units left of one of the two products, and plenty of the other. Find the probability that the $M$ units are enough for every customer in the queue who wants that product. The assessment also asks a variant: the probability that the $M$ units are exactly enough, meaning every unit is sold and no customer who wants the product leaves without it. The specific values of $N$, $M$ and $p$ are generated randomly. ```hint Count the demand Define a random variable for how many customers in the queue want the scarce product, and identify its distribution before thinking about the stock. ``` ### Constraints and Clarifications - $N$ is a positive integer, $M$ is a nonnegative integer, and $p$ is strictly between 0 and 1. - Each customer wants exactly one unit of exactly one product, and that choice does not depend on what is left in stock. - The question is stated with product A as the scarce one. If product B is the scarce one, swap the roles of $p$ and $q$. ### Clarifying Questions - Which of the two products is the one with only $M$ units left? - Does "exactly enough" mean that demand for the product equals $M$, so no unit is left over? - If a customer's product is sold out, do they buy the other product instead, and does that affect the question? - Does each customer buy one unit, or can a customer buy several? ### What a Strong Answer Covers - Modeling the demand for the scarce product as a sum of independent indicators and naming its distribution - The difference between the "enough" and "exactly enough" events, with an expression for each - Efficient and numerically safe evaluation, including the cases $M \ge N$ and $M = 0$ - A reasonableness check, such as a complement or an approximation, for large $N$ ### Follow-up Questions - What is the expected number of customers who want the scarce product but cannot get it? - What is the smallest stock level that serves everyone with probability at least a target level $c$? - Given that the customer in position $j$ of the queue wants the scarce product, what is the probability that they get it?