Poisson Probability Calculator

This Poisson probability calculator works out how likely a given number of events is when those events happen independently at a known average rate. You enter the average number of events expected in your interval, written as lambda, and the count k you are interested in, and the tool returns the probability of exactly k events and the cumulative probability of k or fewer events. The Poisson formula is lambda to the power k, times e to the power minus lambda, divided by k factorial. The cumulative figure adds up that probability for every value from zero up to k. The Poisson model fits situations like the number of calls to a help desk in an hour, the number of faults on a length of cable, customers arriving at a counter, or rare disease cases in a region, as long as the events are independent and the average rate is steady. Analysts, operations planners, students and quality engineers use it to set staffing, judge whether a count is unusually high, and model arrivals and defects. Three tips help you apply it well. First, make sure lambda matches the same interval as the count you care about, so if your rate is per day but you want a weekly count, multiply lambda by seven. Second, the distribution is right skewed for small lambda and becomes more symmetric as lambda grows. Third, the cumulative probability is the natural tool for tail questions, since the chance of more than k events is simply one minus the chance of k or fewer. Always confirm the independence assumption holds before trusting the result.

P(k) = lambda^k * e^-lambda / k!. Estimate only, not financial or tax advice.