These two get confused a lot because they're both about counting things, both give whole-number outcomes, and both show up in the same part of every intro stats course. Here's the actual test to tell them apart.
The core question to ask
Do you know the total number of trials? If yes — binomial. If you're instead counting how many times something happens over a stretch of time or space, with no clear "number of trials" — that's Poisson.
| Binomial | Poisson | |
|---|---|---|
| What it needs | A fixed number of trials (n) and a probability (p) | An average rate of occurrence (λ) over a time/space window |
| Example | Number of heads in 20 coin flips | Number of customers arriving at a store in an hour |
| "Trials"? | Clearly defined and countable | Not really — events just happen at some average rate |
Examples that make it click
Binomial: "Out of 50 emails sent, how many will get opened, if the open rate is 20%?" — you know exactly how many emails (50), so this is binomial.
Poisson: "How many typos will there be on a randomly chosen page of a book, if the average is 2 typos per page?" — there's no defined "number of trials" here, just an average rate. That's Poisson.
The connection between them
Here's the neat part: Poisson distribution is actually what binomial distribution turns into when n gets extremely large and p gets extremely small, but n × p (the mean) stays roughly constant. Think of "typos per page" as technically being made up of a huge number of tiny binomial trials (every character could theoretically be a typo, with a tiny probability each) — but tracking it that way is impossible, so Poisson gives you a cleaner shortcut for that specific situation.
Quick self-check
- Can you clearly state "n trials, each with probability p"? → Binomial. Use the calculator directly.
- Do you only have a rate, like "3 per hour" or "1 per 100 pages," with no natural trial count? → Poisson, which is a different tool than the one on this page.
If your situation genuinely fits the binomial pattern, it's worth double-checking against the four conditions in this beginner's guide before you commit to it.
Try it on the calculator
Plug your own numbers in and see the full step-by-step working instantly.