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Enter the average rate (λ) and number of events (k) to quickly calculate the exact Poisson probability P(X=k) and cumulative probabilities.
λ is the average number of events in a given time/space interval, can be a decimal, and must be ≥ 0. k is the actual number of occurrences you want to calculate, and must be a non-negative integer.
Enter the average rate λ and number of occurrences k to view probabilities.
Overview
Understand what the tool solves, how it works, and the boundaries of its data.
A Poisson model assigns probabilities to counts of events in a defined interval, such as calls per hour or defects per batch. The parameter λ (lambda) is the expected event count for that exact interval; k is the nonnegative whole-number count you want to examine. If an average is stated per hour but your question covers four hours, convert the expected count to the four-hour interval before calculating.
P(X = k) = e−λ λk / k!
P(X ≤ k) = Σi=0k P(X = i)
P(X < k) = P(X ≤ k − 1); P(X > k) = 1 − P(X ≤ k); P(X ≥ k) = 1 − P(X < k)
X denotes the count, e is the natural constant, and k! is the factorial of k. The calculator reports exact, below, at-or-below, above, and at-or-above probabilities, along with the model mean, variance, and standard deviation. For a Poisson model, the mean and variance are λ and the standard deviation is √λ. NIST documents the point-probability and cumulative formulas.
Suppose a service desk averages λ = 4 calls in one hour and you ask for exactly k = 2 calls. Then P(X=2) = e−4 × 42 / 2! ≈ 0.1465, or about 14.65%. “At least 2” includes 2 and gives about 0.9084; “more than 2” excludes 2 and gives about 0.7619. The chart shows point probabilities across a displayed range and highlights the chosen k; it is a visual aid rather than a fitted model from observed data.
Guide
Follow the workflow and verify inputs and outputs with practical examples.
Define one observation interval and express λ as the expected number of events in that interval.
Enter λ as a nonnegative number. Enter k as a nonnegative integer; decimal or negative event counts are invalid.
Read the requested event count probabilities and the mean, variance, and standard deviation. The example button fills λ = 3 and k = 5; reset clears both inputs.
Match the result to your question: “exactly k” is P(X=k), “no more than k” is P(X≤k), “fewer than k” is P(X<k), “at least k” is P(X≥k), and “more than k” is P(X>k).
The calculation is rejected when k exceeds 1,000,000 to limit the cumulative sum. Very small results may be shown in scientific notation.
Use cases
See how the tool fits into real work and everyday tasks.
With a defensible average calls-per-hour figure, compare the chance of exactly a given count with the chance of reaching or exceeding a staffing threshold.
If the batch size and conditions match the interval behind λ, calculate the model probability of a defect count, then assess it alongside process and inspection evidence.
Operations teams can compare “at most” and “at least” request counts for a fixed time window when discussing capacity assumptions.
Q&A
Find concise answers to common questions and confusing cases.
Yes. λ is an expected count and may be fractional, such as 2.5 events per defined interval.
“At least k” includes k and uses P(X≥k). “More than k” starts at k+1 and uses P(X>k).
No. You provide λ. The calculator does not fit a rate from a sample or check whether the model assumptions fit your situation.
Notes
Review scope, result limitations, and important precautions before use.
A calculated probability is conditional on the Poisson model and the λ you enter. Consider whether the count interval is well-defined, the average rate is reasonably stable over it, and events are not strongly dependent or clustered. A changing rate, bursts, seasonality, or linked events may call for a different model. The chart and calculated tails do not diagnose anomalies by themselves. For high-impact decisions, review the data collection and model choice with a qualified statistician or subject-matter expert.
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