Poisson Distribution Calculator

The Poisson distribution models how many events occur in a fixed interval at a known average rate.

How the Math Works

The Poisson Distribution Calculator uses the formula P = e^(-λ) * λ^k / k! to compute the probability of observing exactly k events when the average rate of occurrence is λ. Here, e is Euler's number (~2.71828), λ represents the expected number of events in a fixed interval, k is the specific number of events you're interested in, and k! (k factorial) is the product of all positive integers up to k. This formula assumes events happen independently and at a constant average rate, making it ideal for rare or discrete occurrences in large populations.

Practical Applications

To use this calculator, input the average rate (λ) and the exact number of events (k) you want to analyze. For example, if a call center receives an average of 5 calls per hour (λ=5), the calculator can determine the probability of receiving exactly 3 calls in the next hour (k=3). This is invaluable in fields like operations research, epidemiology, or telecommunications to model arrival processes, predict demand, or assess risk in scenarios where events are counted over time or space.

Day-to-Day Use

In daily life, this tool helps with practical decision-making: estimating the likelihood of traffic jams if an average of 2 accidents occur per week on your commute route, predicting how often a vending machine might malfunction (e.g., 0.5 breakdowns per month), or planning staffing at a retail store based on average customer arrivals. By quantifying these probabilities, you can prepare better for variability, optimize resources, and make data-driven choices in personal or professional contexts.

Worked example

λ 3, k 2 → 22.4%.

FAQ