Poisson Distribution Calculator
Poisson probability P(X = k) for a rate λ with cumulative and tail probabilities — for counts of rare events such as colonies, mutations, reads or arrivals.
The Poisson Distribution Calculator computes P(X = k) = λᵏ·e^(−λ)/k! for a count k when events occur independently at an average rate λ, together with P(X ≤ k), P(X ≥ k) and P(X > k).
Free, runs in your browser. Nothing you enter is uploaded.
How to use it
- Enter λ and k.
- Read the probabilities.
What you enter
- λ, k
What you get
- P(X = k), P(X ≤ k), P(X ≥ k), P(X > k)
Formula
P(X = k) = λᵏ e^(−λ) / k!
The Poisson distribution is the limit of the binomial for many trials with a small success probability; its mean and variance both equal λ.
Worked example
Input: λ = 3.5, k = 2
Result: P(X = 2) = 0.1850, P(X ≤ 2) = 0.3208
With an average of 3.5 colonies per plate, about 18.5 % of plates show exactly two.
Questions
When is a Poisson model appropriate?
For counts of independent rare events in fixed time, area or volume: colonies per plate, mutations per genome, partitions per droplet in dPCR, cells per field.
Limitations
- Assumes events are independent and the rate is constant across the observation window.
References
- Huggett JF et al. (2013). The digital MIQE guidelines. Clin Chem 59:892-902.
Related reading
- Poisson Statistics in Digital PCR: Absolute Copy Numbers from Partition Data
- How to Choose the Right Probability Distribution for Biological Data
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All free tools · By Dr. Zubair Khalid, DVM, MS, PhD.