Poisson probability is one of 7 distributions formulas in the probability section of this library, and it is used at university level.
Why poisson probability works
It is the shape a binomial takes when there are an enormous number of opportunities each with a tiny chance of an event, with the average count held steady. Everything except that average washes out in the limit, which is why the expected count is the only input this distribution needs.
What each symbol means
$\lambda>0$ is expected count in the interval.
Poisson probability: when it holds
Events should occur independently with approximately constant rate.
When it stops applying
It assumes events happen independently at a steady rate. Arrivals that clump, such as buses bunching in traffic or goals that change how hard a team attacks, break independence, and then the observed variance exceeds the mean instead of matching it.
Poisson probability: a worked example
If $\lambda=2$, $P(X=0)=e^{-2}$.
The mistake to avoid
What people do: Using a rate quoted for one interval length while asking about a different one.
Why it goes wrong: A rate of 3 arrivals per hour gives an average of 1 over a twenty-minute window, not 3. Leaving the hourly figure in place triples every probability in the calculation.
Do this instead: Scale the rate to match the interval in the question before substituting, multiplying by the fraction of the period you are actually asking about.
Poisson probability: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\lambda>0$ is expected count in the interval.
- Check the conditions before substituting. Events should occur independently with approximately constant rate.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Distributions
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where poisson probability comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about poisson probability
Is the mean really equal to the variance?
Yes, both equal the rate parameter, which is a strong and testable claim. If your data show a variance clearly larger than the mean, the Poisson model is the wrong one.
How do I compute the chance of at least one event?
Subtract the chance of none from 1, which is 1 minus e to the minus rate. With an average of 2, that gives about 0.8647.
Can the average count be a decimal?
Yes, and usually it is. An average of 0.7 accidents per week is perfectly meaningful even though only whole numbers of accidents ever occur.
When can Poisson stand in for a binomial?
When the trial count is large and the success probability is small. With 1000 trials at probability 0.002 the binomial gives 0.18063 for exactly three events and Poisson gives 0.18045.