Which one is it?
Before any formula, decide what is being counted. That single question picks the distribution.
Poisson
“How many, in this window?”
- Counts events, not trials — there are no trials to count.
- Needs a fixed interval: an hour, a page, a square metre.
- Events arrive one at a time, independently, at a constant average rate.
- X can be 0, and has no upper limit.
- One parameter: λ, the mean count per interval.
Geometric
“How long until the first one?”
- Counts trials, and you stop at the first success.
- Trials are independent, each with the same success chance p.
- X is at least 1 — the successful trial is counted — and has no upper limit.
- The number of trials is not decided in advance; that is what makes it not binomial.
- One parameter: p.
And for contrast, the binomial from the previous section answers a third question — “in a fixed number of trials n, how many succeeded?” If n is pinned down in advance, it is binomial; if you are counting events in a window it is Poisson; if you are counting trials until the first success it is geometric.