Exactly k, at most k, at least k or between a and b events when all you know is the average rate λ, with the whole Poisson distribution table and the mean, SD and most likely count.
Poisson Distribution Calculator - Exactly, At Most, At Least
Exactly k, at most k, at least k or between a and b events when all you know is the average rate λ, with the whole Poisson distribution table and the mean, SD and most likely count.
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Poisson probabilities from nothing but an average
How likely are exactly 6 calls in the next hour when a help line averages 4 an hour? The Poisson distribution puts it at 10.42%, and this calculator returns that number together with the "at most" and "at least" versions, a range from a to b and a table of the whole distribution. Two inputs are enough: the average count λ for one period and the count k you are asking about. When the average is per hour and the question covers three hours, enter the number of periods and λ is scaled for you.
Using the Poisson calculator, field by field
- Which probability - exactly k, at most k, at least k, or between a and b inclusive. "None at all" is exactly 0, "at least one" is at least 1.
- Average number of events λ - the mean count in one period or area, such as 2.7 goals per game or 0.5 typos per page. Decimals are fine, typed with a dot.
- Number of periods - optional. The average is per page and you ask about a 10-page chapter? Enter 10 and λ grows to 5.
- Number of events k (and the upper bound b for a range) - always a whole number, because events are counted, not measured.
- Read the result - a large tile with the answer to your question, six smaller tiles with the related probabilities, the mean, the standard deviation and the most likely count, and a distribution table with your k highlighted.
The formula, and three problems worked to the end
The probability mass function in the NIST/SEMATECH e-Handbook of Statistical Methods is P(X = k) = λk e-λ / k! for k = 0, 1, 2 and so on. "At most k" adds the bars from 0 to k, "at least k" is 1 minus the sum up to k - 1, and both the mean and the variance equal λ. The calculator works with logarithms of the factorials, so it keeps full precision up to λ = 1,000, where a handheld calculator already overflows on 1000!.
Poisson or binomial? A side-by-side comparison
Both distributions count events, but they answer different questions. The binomial needs a known number of trials and a probability per trial. The Poisson needs only the average over a period, with no trials to count.
| Feature | Poisson | Binomial | Pick it when |
|---|---|---|---|
| Inputs | average λ per period | trials n and probability p | you only know an average: Poisson |
| Largest possible k | no upper limit | n | the count is capped by the trials: binomial |
| Mean | λ | n × p | - |
| Variance | λ, equal to the mean | n × p × (1 - p), below the mean | variance far above the mean: neither fits well |
| Typical use | calls per hour, goals per game, typos per page, failures per month | hits in 10 shots, defects in a batch of 50, correct answers on a quiz | events "over time": Poisson; "per trial": binomial |
With many trials and a small p the two give almost the same answer, so a Poisson with λ = n × p can stand in for the binomial. The NIST e-Handbook gives the rule of thumb: n of at least 20 and p of at most 0.05, with an excellent approximation when n is 100 or more and n × p is at most 10. Its own check uses n = 200 and p = 0.025: exactly 3 successes come to 0.1399995 under the binomial and 0.1403739 under the Poisson with λ = 5.
The table below repeats that check for a batch of 100 items, each defective with probability 2%. No difference exceeds three tenths of a percentage point.
| Defective items k | Binomial (exact) | Poisson λ = 2 | Difference |
|---|---|---|---|
| 0 | 13.26% | 13.53% | +0.27 |
| 1 | 27.07% | 27.07% | 0.00 |
| 2 | 27.34% | 27.07% | -0.27 |
| 3 | 18.23% | 18.04% | -0.18 |
| 4 | 9.02% | 9.02% | 0.00 |
| 5 | 3.53% | 3.61% | +0.07 |
| 6 | 1.14% | 1.20% | +0.06 |
Poisson probability between two numbers
A range question needs two cumulative sums and a subtraction, which is where hand calculations usually slip by one. At λ = 3, "from 2 to 5 events" is P(X ≤ 5) - P(X ≤ 1) = 91.61% - 19.91% = 71.69%. Note the 1, not the 2: the lower bound itself belongs to the range, so the sum that is taken away stops one step below it. The calculator does this for you when you choose "from a to b", and the table under the result highlights every bar inside the range.
The same idea covers planning questions. A help line that averages 4 calls an hour sees λ = 32 over an 8-hour shift. A shift with at most 25 calls happens 12.28% of the time, and one with 40 or more 9.560% of the time, so roughly one shift in ten needs extra hands.
Where Poisson answers go wrong
Poisson distribution questions
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