Enter an average rate (λ) and a count to get the chance of exactly that many events, or that many and fewer, or that many and more — with a probability chart, all client-side.
Enter the average rate λ and a count x, choose the probability type, and press Calculate.
Defects arrive at an average rate of 3 per unit. What is the chance of seeing exactly 5? Enter the lab to work it out.
Type the average rate you expect — the typical number of events in your interval — and the count you're interested in. Choose whether you want the exact chance, or the chance of "that many or fewer" / "that many or more".
The calculator works out the exact chance of seeing your chosen count, based on the average rate you entered. It uses the Poisson formula — a special tool built for counting rare, scattered events. It stays accurate even for large numbers, and automatically adds up probabilities for 'that many or fewer' and 'that many or more'.
The chart shows the chance of every possible count around your target value, with your chosen count highlighted. The average and the spread both depend on the rate you entered — higher rate means more spread.
The Poisson distribution models the count of rare, independent events over a fixed interval of time, area, or volume — defects per metre, calls per hour, breakdowns per month. With a single parameter λ (the average rate) you can work out the chance of any number of events occurring.
The Poisson distribution gives the probability of observing a given number of independent events in a fixed interval when they occur at a constant average rate λ. It applies when events are rare relative to the opportunities for them to happen, occur independently, and cannot coincide at exactly the same instant. Examples include defects per unit, arrivals per hour, or failures per month.
The Poisson model describes how often rare, independent events happen when you only know the long-run average rate. Give it that average and a count, and it returns the probability of seeing exactly that many — plus the cumulative chances of 'that many or fewer' and 'that many or more'. For example, with an average of 3 defects you can ask how likely 5 or more would be. A handy property of a Poisson process is that its average and its variance are the same number.
A production line averages 3 defects per batch — that's completely normal for this process. The team wants to know: how worried should they be if the next batch has 5 defects?
Entering those numbers gives roughly an 18.5% chance — about 1 in 5 batches will hit 5 or more defects just by chance, even when everything is running fine.
What to do with this: Don't panic at 5 defects — it's not unusual. But 8 or 9 defects in a batch? That's only a 1–2% chance and worth a proper investigation. Set your alert threshold based on what's genuinely abnormal, not just higher than average.
Counts of defects, breakdowns, complaints, and incidents drive quality and reliability decisions. The Poisson distribution turns an average rate into the probability of specific outcomes, so you can set realistic thresholds, plan capacity, and judge whether an unusually high count is genuine cause for concern or just normal variation.
Reach for the Poisson distribution whenever you are counting events over a fixed exposure — per item, per hour, per area — rather than counting successes out of a fixed number of trials. It underpins C and U attribute control charts, defect-per-unit metrics, and queueing models used in the Measure and Analyse phases of DMAIC.
Defects-per-unit and defects-per-opportunity thinking is fundamentally Poisson. Understanding the distribution helps you interpret DPU and DPMO, choose the right attribute control chart, and reason about how likely a spike in defect counts really is before launching a special-cause investigation.
Use the probability to set sensible thresholds and acceptance rules — for example, the count beyond which you would investigate a special cause. If you are monitoring counts over time, move the analysis onto a C or U attribute control chart so you can see shifts and trends, not just one-off probabilities. If your real data shows much more spread than λ would predict (variance well above the mean), consider a negative binomial model instead, and revisit whether the events are truly independent.
Combine Poisson probabilities with attribute control charts and DPMO metrics to monitor and improve defect rates.
Use the Poisson distribution when you are counting how many times an event happens over a fixed interval of time, area, or volume, and there is no fixed number of trials — for example defects per metre of cable, calls per hour, or accidents per month. Use the Binomial distribution when you have a fixed number of independent trials, each with a pass/fail outcome and a constant probability — for example the number of defective units in a batch of 200. As a rule of thumb, when the number of trials n is large and the probability p is small, the Binomial is well approximated by a Poisson with λ = np.
Lambda (λ) is the average number of events you expect in the interval you are studying — the mean rate. In a Poisson distribution the mean and the variance are both equal to λ, so the standard deviation is √λ. For example, if a process averages 3 defects per batch then λ = 3, the variance is 3, and the standard deviation is about 1.73. You can estimate λ from data by taking the average count over many equal intervals.
The exact probability P(X = x) is the chance of seeing exactly x events. The cumulative probability P(X ≤ x) adds up the chances of 0, 1, 2, … up to x events, answering 'x or fewer'. P(X ≥ x) is the complement that answers 'x or more', equal to 1 − P(X ≤ x−1). Use the exact form for a specific count, and the cumulative forms when you care about a threshold, such as the chance of two or fewer breakdowns this week.
The Poisson distribution models counts of rare, independent events: defects per square metre of sheet metal, customer arrivals per hour at a help desk, machine breakdowns per month, typing errors per page, or safety incidents per quarter. It applies whenever events occur at a roughly constant average rate, independently of one another, and two events cannot occur at exactly the same instant. It is widely used in Six Sigma for defect-rate and reliability work.
Poisson probabilities turn average event rates into actionable thresholds. Structured training shows you how to monitor and reduce defect rates over time.
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