In determining which Distribution to use in analyzing Discrete (Count) data, we need to know whether we are interested in Occurrences or Units.
Let's say we are inspecting shirts at the end of the manufacturing line. We may be interested in the number of defective Units – shirts, because any defective shirt is likely to be rejected by our customer. However, one defective shirt can contain more than one defect. So, we are also interested in the Count of individual defects – the Occurrences – because that tells us how much of a quality problem we have in our manufacturing process.
For example, if 1 shirt has 3 defects, that would be 3 Occurrences of a defect, but only 1 Unit counted as defective.
We would use the Poisson Distribution in analyzing Probabilities of Occurrences of defects. To analyze the Probability of Units, we could use the Binomial or the Hypergeometric Distribution.
There is an article in the book focusing on the Poisson Distribution. There is also a video, on my YouTube channel, Statistics from A to Z.
Andrew A. (Andy) Jawlik is the author of the book, Statistics from A to Z -- Confusing Concepts Clarified, published by Wiley.