Poisson Distribution Calculator

Poisson Distribution Probability Calculator

This free Poisson distribution calculator finds the probability of a specific number of events occurring within a fixed interval of time, distance, area, or volume, given only the average rate at which those events happen. It is one of the most widely used discrete probability distributions in statistics, operations research, and everyday planning — anywhere you need to answer a question of the form 'how likely is it that exactly x events happen, given that on average λ events happen per interval?'

Unlike the binomial distribution, which needs a fixed number of trials (n) and a per-trial success probability (p), the Poisson distribution only needs one parameter: the average rate of occurrence, usually written as λ (lambda). This makes it the natural choice whenever events happen independently and continuously over time or space rather than in a fixed set of discrete trials — for example, the number of customer arrivals at a store per hour, the number of emails received per minute, the number of website errors per day, or the number of typos per page.

When to Use the Poisson Distribution

The Poisson distribution applies whenever three conditions hold: events occur independently of one another, the average rate of occurrence (λ) is constant over the interval being studied, and two events essentially cannot occur at exactly the same instant. Classic real-world examples include the number of phone calls a call center receives per hour, the number of car accidents at an intersection per month, the number of radioactive decay events per second, the number of typing errors per page, the number of customer arrivals at a bank per interval, and the number of defects found per batch in quality control. Anywhere you're counting how many times a rare, independent event happens in a fixed window, the Poisson distribution is the right probability model.

Poisson Probability Formula

The probability of observing exactly x events in an interval, given an average rate of λ events per interval, is given by the Poisson probability mass function (PMF):

P(X = x) = (e^−λ · λ^x) / x!

Here, e is Euler's number (approximately 2.71828), λ is the average number of events per interval, x is the actual number of events whose probability you want to find, and x! is the factorial of x. The term e^−λ decays as λ grows, λ^x grows with x, and dividing by x! keeps the whole expression a valid probability between 0 and 1. This calculator computes the formula in log-space internally so that it stays numerically accurate even for large λ or large x, where λ^x and x! individually would overflow standard floating-point numbers.

To find the probability of a range of outcomes instead of one exact count, the calculator uses the Poisson cumulative distribution function (CDF): P(X ≤ x) = Σ P(X = i) for i = 0 to x. From there, P(X ≥ x) = 1 − P(X ≤ x − 1), and P(a ≤ X ≤ b) sums the PMF from i = a to i = b — exactly the same logic used for the binomial distribution, just with the Poisson PMF instead.

Mean, Variance, and Standard Deviation

One of the most distinctive properties of the Poisson distribution is that its mean and variance are always equal, both equal to λ. So the mean number of events per interval is μ = λ, the variance is σ² = λ, and the standard deviation is σ = √λ. This calculator computes all three automatically alongside your probability result, which is useful for judging how unusual a particular outcome is relative to the average — an outcome several standard deviations away from λ is comparatively rare.

How to Use This Calculator

Choose what you want to find: the probability of exactly x events, at most x events, at least x events, or a probability between two values. Enter the average rate (λ) of events per interval, and the number of events (x) you're interested in. The calculator instantly returns the exact probability, its percentage form, the mean, variance, and standard deviation, along with a bar chart showing the full probability distribution around λ.

• P(X = x) — exact probability, useful for questions like 'what's the chance exactly 5 customers arrive in the next hour?'
• P(X ≤ x) — cumulative probability, useful for 'what's the chance of at most 2 machine breakdowns this month?'
• P(X ≥ x) — probability of x or more, useful for 'what's the chance of at least 10 support tickets today?'
• P(a ≤ X ≤ b) — probability within a range, useful for capacity-planning or staffing questions.

Worked Example

A call center receives an average of 4 calls every 10 minutes (λ = 4). What is the probability of receiving exactly 6 calls in the next 10 minutes? Using the formula: P(X = 6) = (e^−4 · 4^6) / 6! = (0.0183 × 4096) / 720 ≈ 0.1042, or about 10.42%. The mean number of calls per 10-minute interval is μ = 4, the variance is σ² = 4, and the standard deviation is σ = 2. Enter λ = 4 and x = 6 into this calculator to see the same result instantly, along with the complete probability distribution charted around the average rate.

Poisson vs. Binomial vs. Normal Distribution

All three of these are common probability distributions, but they model different situations. The binomial distribution needs a fixed number of trials (n) with a constant success probability (p) per trial — use the binomial distribution calculator for that. The Poisson distribution needs only an average rate (λ) over a continuous interval, with no fixed upper limit on how many events could occur — use this calculator whenever you're counting rare, independent events over time or space rather than counting successes out of a known number of attempts. The normal distribution, meanwhile, models continuous measurements (like height or exam scores) rather than event counts — for large λ, the Poisson distribution starts to closely resemble a normal distribution, which is why the normal distribution calculator can sometimes be used as an approximation for large-λ Poisson problems.