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Poisson distribution calculator

The probability of exactly k events when they happen at an average rate λ per interval, independently — P(X=k) = e^(−λ)·λ^k / k!. It returns the exact probability, the cumulative P(X≤k), the tail P(X≥k), and reminds you that the mean and variance both equal λ.

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Enter Average rate λ, Number of events k and the Poisson distribution calculator works out P(X = k), P(X ≤ k), P(X ≥ k), Mean = variance = λ straight away. For instance, with Average rate λ = 5 and Number of events k = 3 it returns P(X = k) = 0.14, P(X ≤ k) = 0.265 and P(X ≥ k) = 0.875.

How to use it

  1. Enter your values: Average rate λ, Number of events k.
  2. Read the result instantly: P(X = k), P(X ≤ k), P(X ≥ k), Mean = variance = λ.

Frequently asked questions

What does the Poisson distribution calculator actually compute?

It takes Average rate λ and Number of events k and derives P(X = k), P(X ≤ k), P(X ≥ k) and Mean = variance = λ from them. The calculation is live as you type, so the result updates on every change.

What information do I need to provide?

2 values: Average rate λ and Number of events k. Nothing else is required — no account, no file upload.

Can you show a worked example?

With Average rate λ = 5 and Number of events k = 3, the calculator returns P(X = k) = 0.14, P(X ≤ k) = 0.265 and P(X ≥ k) = 0.875. Those figures come from running this exact tool, so you can reproduce them by entering the same values.

What happens if I enter larger values?

It moves a lot. Using Average rate λ = 5.5 and Number of events k = 6 instead, P(X = k) goes from 0.14 to 0.157 — which is why it is worth testing a few scenarios rather than trusting a single figure.

What does it give for smaller values?

Scaled down to Average rate λ = 4.5 and Number of events k = 2, P(X = k) comes out at 0.112. The relationship is worth checking at both ends before you rely on a single result.

When would I actually use this?

Summarising a dataset before drawing conclusions from it, checking whether a difference between two groups is real, and putting an interval around an estimate.

What is the most common mistake?

Reading a p-value as the probability that the hypothesis is wrong. It is the probability of seeing data at least this extreme if the null were true — a different statement, and a much weaker one.

What is the difference between the Poisson distribution calculator and the Beta Distribution Calculator?

This one returns P(X = k) and P(X ≤ k); the Beta Distribution Calculator returns Result. That is the whole difference — open the one whose figure you need.

Is there a tool for the next step?

Exponential Distribution Calculator is the closest one after this: Compute exponential-distribution probabilities for waiting times from the rate λ, including P(X≤x), P(X>x) and a range.

What else is worth having open alongside it?

Gaussian Distribution Generator and Geometric Distribution Calculator — they come up in the same task often enough to be worth a second tab.

Further reading

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ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerWhat Is a Z-Score? Standard Deviations Above the MeanA z-score is z = (x − μ) / σ — how many standard deviations a value sits from the mean. Learn how to compute it, read the standard normal curve, and turn it into a percentile.ExplainerMean, Median and Mode: What's the Difference?Mean, median and mode all describe the 'middle' of your data — but answer different questions. Here's what each means, with examples and when to use which.ExplainerGeometric vs Arithmetic Mean: Which to UseThe arithmetic mean adds and divides; the geometric mean multiplies and takes a root. Learn which fits growth rates, ratios, and averages.How-toHow to Calculate Standard Deviation (Step by Step)Standard deviation measures how spread out your data is. Here's the formula, a worked example, and the difference between population and sample.ExplainerProbability Basics, With ExamplesProbability measures how likely something is, from 0 to 1. Here's the basic idea, how to combine events with AND and OR, and the complement.