Poisson Probability Calculator
Calculate the Poisson probability P(X = k) for a given rate λ and count k.
How to use this tool
- Enter rate (λ) and events (k) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your p(x = k) and the full breakdown beneath it.
P(X = k) = e^(−λ) × λ^k / k!. Useful for rare events: phone calls, defects per unit, arrivals per hour.
Formula
P(X = k) = (e−λ × λk) / k!
Computed in log-space for numerical stability: ln P = −λ + k × ln λ − ln(k!), then exponentiated.
How it works
This calculator gives the probability of observing exactly k events in a fixed interval when events occur independently at a constant average rate λ (lambda).
The Poisson model is appropriate when events are rare relative to the number of opportunities, occurrences are independent, and the rate is constant over the observation window. It is widely used in queuing, reliability engineering, and epidemiology.
Worked example
Worked example
- Inputs: rate λ = 3.0 events per interval, k = 2 events.
- Compute e−λ: e−3 ≈ 0.049787.
- Compute λk / k!: 3² / 2! = 9 / 2 = 4.5.
- P(X = 2) = 0.049787 × 4.5 ≈ 0.2240.
P(X = 2) = 0.224
Key terms
- Poisson distribution
- A discrete probability distribution giving the probability of a number of events occurring in a fixed time or space when the average rate λ is known and events are independent.
- Rate parameter (λ)
- The expected (average) number of events in the observation interval; also equals the mean and variance of the Poisson distribution.
- Independence
- The assumption that the occurrence of one event does not influence the likelihood of another event in the same interval.
- Rare-event approximation
- The Poisson distribution approximates the binomial distribution when n is large and p is small, with λ ≈ n × p.
- Cumulative Poisson probability
- P(X ≤ k) is found by summing individual Poisson probabilities P(X = 0) + P(X = 1) + … + P(X = k).
Frequently asked questions
- When should I use the Poisson distribution?
- Use Poisson when counting events in a fixed interval of time or space, where events occur independently at a known average rate λ. Examples: emails per hour, accidents per week, typos per page.