Calculate exact, at-most, at-least, greater-than, and between-count Poisson probabilities from λ, k, and interval scaling, with mean, variance.
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Poisson distribution calculator Calculate exact, at-most, at-least, greater-than, and between-count Poisson probabilities from an average rate λ. Use the interval multiplier when your rate is per hour, day, item, or area but your question uses a different interval.
Quick examples
Start with arrivals, defects, a scaled time interval, or a range probability, then adjust λ, the event count, and the query type to match your own Poisson probability question.
Average 4 calls per hour, asking for exactly 3 calls in the next hour.
Probability question
Result
0.195367
P(X = 3) = 0.195367 with effective λ = 4. That result is plausible but not dominant under the stated model.
λ already matches the tested interval, so effective λ is 4.
0.195367
P(X = 3)
0.195367
P(X = 3)
0.43347
P(X ≤ 3)
0.761897
P(X ≥ 3)
4
Mean / effective λ
4
Variance
2
Standard deviation
P(X = 3) — exact probability
0.195367
P(X ≤ 3) — at most 3 events
0.43347
P(X ≥ 3) — at least 3 events
0.761897
P(X > 3) — more than 3 events
0.56653
P(X < 3) — fewer than 3 events
0.238103
Event count
Exact P(X = k)
At most P(X ≤ k)
At least P(X ≥ k)
0
0.018316
0.018316
1
1
0.073263
0.091578
0.981684
2
0.146525
0.238103
0.908422
3
0.195367
0.43347
0.761897
4
0.195367
0.628837
0.56653
5
0.156293
0.78513
0.371163
6
0.104196
0.889326
0.21487
How to read the result
Use this tail and exact probability alongside the model assumptions: independent events, a constant average rate, and the same time, area, or volume interval for λ and the event count.
P(X = 3) = e^(-4) × 4^3 / 3! = 0.195367
Model reminder Use a Poisson distribution when events occur independently at a constant average rate. If variance is much larger than the mean, the Poisson model may understate tail risk.
Poisson distribution calculator guide: exact, cumulative, range
The Poisson distribution models the number of events occurring in a fixed interval of time, space, area, or volume when the average rate λ is known.
When to use the Poisson distribution
The Poisson distribution is appropriate when events occur independently, at a constant average rate, and the probability of two events at exactly the same instant is negligible. Typical applications include: calls arriving at a call centre per hour, website visits per minute, bacteria colonies on a petri dish, defects per unit area.
The distribution is characterised by a single parameter λ (lambda), which equals both the mean and the variance.
This makes a Poisson probability calculator especially useful for count data where the possible number of events is open-ended rather than limited by a fixed number of trials. If you are counting successes out of a known sample size, a binomial distribution calculator is usually the better model.
Choose the probability question before interpreting the answer
A strong Poisson calculator should answer more than P(X = k). Exact probability answers questions such as "what is the probability of exactly 3 calls?" Cumulative probability answers "at most 3 calls" with P(X ≤ 3), upper-tail probability answers "at least 3 calls" with P(X ≥ 3), and strict tail probability answers "more than 3 calls" with P(X > 3).
Range probability is often the practical planning view. P(2 ≤ X ≤ 6), for example, tells you the chance that the count stays inside an acceptable band instead of only matching one exact count. That is useful in staffing, quality control, inventory arrivals, support queues, and other operations work.
Poisson vs binomial
The Poisson distribution is the limit of the binomial when n → ∞ and p → 0 with n·p = λ constant. In practice, use Poisson when counting events in a continuum (time, area, volume) with a very large or unbounded number of possible events.
Use binomial when there is a fixed number of trials n with clear success/failure outcomes.
The Poisson approximation to the binomial is most natural when events are rare, trials are numerous, and the expected count n·p is the rate you want to model. If p is not small or trials are not independent, keep the binomial model instead of forcing a Poisson model.
Make sure λ matches the interval you actually care about
A common source of mistakes is using a rate from one interval to answer a question about another interval without converting λ first. If the average rate is 4 calls per hour and you want the probability for 30 minutes, the relevant Poisson rate is λ = 2 for that half-hour interval, not λ = 4.
This is one reason Poisson pages that rank well usually show both the formula and a real interpretation step. The calculation is straightforward once λ matches the same time, area, or volume interval as the event count being tested.
The interval multiplier in this calculator makes that conversion visible. Use 0.5 for half the base interval, 0.25 for a quarter interval, 2 for double the interval, or 1 when your λ already matches the interval in the question.
Worked example: probability of exactly 3 calls when λ = 4
Suppose a call centre receives an average of 4 calls per hour and you want the probability of seeing exactly 3 calls in the next hour. In Poisson notation that is P(X = 3) with λ = 4. The exact probability works out to about 0.195, which means there is roughly a 19.5% chance of seeing exactly 3 calls in that hour.
The same setup also gives a cumulative view and an upper-tail view. P(X ≤ 3) tells you the chance of seeing 3 or fewer calls, while P(X ≥ 3) tells you the chance of seeing at least 3 calls. Those are often the more practical questions in staffing, defects, or arrivals work because they show whether the observed count sits in the common part of the distribution or in the tail.
If the practical question is a range, switch to between-count mode. With λ = 4, P(2 ≤ X ≤ 6) is the chance that the hour stays within a moderate operating band rather than producing a very quiet or unusually busy interval.
How to read tail probabilities
Tail probabilities help you decide whether an observed count is ordinary under the model or unusually high or low. A small P(X ≥ k) means counts at least that large are rare if the stated λ and model assumptions are right. A small P(X ≤ k) means counts that low are rare.
Tail probability is not proof that the model is wrong. It is a signal to check the assumptions: whether the rate stayed constant, whether events were independent, whether multiple event types were mixed together, and whether the interval conversion was correct.
When the Poisson model may be weak
Poisson assumes the mean and variance are both λ. Real count data often has overdispersion, where variance is larger than the mean because events cluster, rates change over time, or observations come from mixed populations.
If variance is much larger than the mean, a negative binomial model, a time-varying rate model, or a simulation may be more appropriate. If events have a hard maximum because there is a fixed number of opportunities, use a binomial or hypergeometric model instead.
Frequently asked questions
What does λ (lambda) represent?
λ is the average rate of events in the interval of interest. For example, if an average of 4 customers arrive per hour, λ = 4. If you observe over a different interval, scale λ accordingly (e.g., λ = 8 for two hours).
Can λ be non-integer?
Yes. λ is a rate and can be any positive real number. k (the number of events) must be a non-negative integer.
Should λ match the exact time interval I am testing?
Yes. If your average rate is expressed per hour but the question is about 15 minutes or 2 hours, convert λ to that same interval first. The event count k and the rate λ must refer to the same interval.
What is the difference between P(X ≥ k) and P(X > k)?
P(X ≥ k) includes k itself, so it means at least k events. P(X > k) excludes k, so it means more than k events. For discrete count data those are different probabilities.
How do I calculate between two Poisson counts?
For an inclusive range, subtract cumulative probabilities: P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1). The calculator's between-count mode does this directly and highlights the selected counts in the probability table.
When does the Poisson approximate to the normal?
When λ is fairly large, the Poisson distribution can be approximated by a normal distribution with mean λ and variance λ. For smaller λ values, the exact Poisson calculation is the better choice.
Can k be larger than λ?
Yes. λ is the average event rate, not a maximum. Individual intervals can produce counts above or below λ, and the Poisson formula handles any non-negative integer k.
Why does the calculator reject decimal event counts?
The Poisson distribution is a count distribution, so the event count must be 0, 1, 2, 3, and so on. λ can be decimal, but k, lower count, and upper count must be whole non-negative counts.
What should I use if variance is larger than the mean?
Large overdispersion suggests the Poisson assumptions may be too strict. Check for changing rates, clustering, or mixed populations; a negative binomial model is often a better count-data model when variance is greater than the mean.
