Poisson Distribution Calculator

P(X = k): The probability of observing exactly k events in the window.

P(X ≤ k) and P(X ≥ k): Cumulative probabilities for planning questions such as "what are the odds of two or more failures this month?"

Distribution summary: The mean λ, the variance λ (a defining Poisson property), and the standard deviation √λ.

The Poisson distribution describes counts of events that happen randomly but at a stable average rate. Three conditions need to hold:

• Independence: One event occurring does not make another more or less likely. A machine breakdown that triggers a cascade of further breakdowns violates this.
• Constant rate: The average number of events per unit of time or space stays the same across the window. Lunchtime rushes at a restaurant mean the hourly rate is not constant across the day.
• No simultaneous events: Events arrive one at a time. Group arrivals — a bus unloading thirty customers at once — should be modeled as batch arrivals instead.

Unlike the binomial distribution, there is no fixed number of trials: the count in a Poisson process has no hard upper limit, only rapidly shrinking probabilities for large values.

The calculator evaluates the probability mass function directly:

The equality of mean and variance is the signature of the Poisson distribution. If your observed counts have a variance far larger than their mean — overdispersion — a negative binomial model usually fits better.

A convenient property of Poisson processes is that the rate scales linearly with the size of the window. If a website averages 12 signups per hour, then over a 15-minute window λ = 12 × (15/60) = 3, and over an 8-hour shift λ = 96. Always rescale λ before entering it, rather than rescaling the answer afterward.

The same logic works for space instead of time: flaws per square meter of fabric, potholes per mile of road, or typos per page. For example, with an average of 1 typo per page, the probability that a given page is typo-free is P(X = 0) = e^(−1) ≈ 0.367879 — about 37%.

Poisson counts appear across operations, science, and risk management.

• Call centers and queues: Staffing models start from the distribution of arrivals per interval to predict wait times.
• Reliability engineering: The number of component failures in a maintenance period drives spare-part inventory decisions.
• Insurance: Claim counts per policy year are the classic actuarial application, feeding directly into premium calculations.
• Epidemiology and biology: Case counts in a region, mutations per genome segment, and radioactive decays per second all follow Poisson models when the underlying rate is stable.

A bakery receives an average of λ = 4 custom cake orders per day. What is the probability of receiving exactly 2 orders tomorrow?

1. Write the formula: P(X = 2) = (4² × e⁻⁴) / 2!
2. Evaluate each piece: 4² = 16, e⁻⁴ ≈ 0.018316, and 2! = 2.
3. Combine: P(X = 2) = (16 × 0.018316) / 2 ≈ 0.146525, a 14.7% chance.

The cumulative outputs answer the planning questions. P(X ≤ 2) = P(0) + P(1) + P(2) ≈ 0.018316 + 0.073263 + 0.146525 ≈ 0.238103, so there is roughly a 24% chance of a slow day with two orders or fewer. Conversely, P(X ≥ 2) = 1 − P(X ≤ 1) = 1 − 0.091578 ≈ 0.908422 — the bakery can count on at least two orders about 91% of the time.

The distribution summary is immediate: mean = 4 orders, variance = 4, and standard deviation √4 = 2. Entering λ = 4 and k = 2 in the calculator reproduces all of these values.