How to use this calculator
- Enter or choose average number of events (λ).
- Enter or choose number of events (k).
- The p(X = k) updates as you type. Use Copy result or Share to keep it.
Method
P(X = k) = e^(−λ)·λᵏ ÷ k!; P(X ≤ k) = Q(k + 1, λ); P(X ≥ k) = P(k, λ)
- λ
- average number of events in the interval
- k
- number of events
- P, Q
- regularised lower and upper incomplete gamma functions (exact cumulative sums)
Worked example
- P(X = 3) = e⁻² × 2³ ÷ 3! = 0.135335 × 8 ÷ 6 = 0.180447.
- P(X ≤ 3) = e⁻² × (1 + 2 + 2 + 4/3) = 0.857123.
- P(X ≥ 3) = 1 − P(X ≤ 2) = 1 − e⁻² × 5 = 0.323324.
If a help desk averages 2 calls an hour, the chance of exactly 3 calls in an hour is 18.0% and of 3 or more is 32.3%.
What the result means
Use “at least” for capacity questions (how likely is a busy hour) and “at most” for service levels (how often can we handle everything). Variance equals the mean in a Poisson process; if your real counts vary much more than that, the data are over-dispersed and a negative binomial model fits better.
Assumptions and limits
- Events occur independently, one at a time, at a constant average rate over the interval.
- Rates that change through the day (rush hours, seasonality) break the constant-rate assumption; split the interval instead.
Questions
How do I change the interval?
Scale λ. If you average 2 calls per hour, use λ = 1 for half an hour and λ = 16 for an 8-hour shift.
How does Poisson relate to the binomial?
For many trials with a small success probability, binomial(n, p) is close to Poisson with λ = np.