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.

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?