About the Poisson Distribution Calculator
This Poisson distribution calculator gives the probability of seeing a certain number of events in a fixed interval when events happen independently at a known average rate. Enter the mean number of events λ (lambda) and the count k, and it returns the exact probability P(X = k) together with the cumulative probabilities P(X < k), P(X ≤ k), P(X > k) and P(X ≥ k).
Typical uses are counting arrivals and rare events: customers per hour at a counter, calls per minute to a help desk, typos per page, defects per batch, goals in a football match or website sign-ups per day. Students use it for probability and statistics homework; analysts use it for staffing, capacity and quality-control questions.
The Poisson model assumes events occur one at a time, independently, at a constant average rate over the interval. Make sure λ is expressed for the same interval as k — if you average 12 calls an hour, λ for a 15-minute window is 3.
With the default inputs, the p(x = k) is 0.100819. Change any value above to recalculate instantly.
How to use the poisson distribution calculator
- 1Enter the average number of events λ for your interval.
- 2Enter the number of events k you are interested in.
- 3Read P(X = k) for "exactly k" and the cumulative results for "at most" or "at least".
- 4Use the chart and table to see the whole distribution.
Formula and method
The Poisson distribution gives the probability of exactly k events when events occur independently at a constant average rate λ per interval. The probability mass function is λᵏe^(−λ)/k!. Cumulative probabilities add these terms from 0 up to k; "more than" and "at least" probabilities are their complements, 1 − P(X ≤ k) and 1 − P(X < k).
To stay accurate for large λ and k, the calculator works with logarithms, computing exp(k·ln λ − λ − ln k!) instead of the raw powers and factorials, which would overflow. The mean and the variance of a Poisson distribution both equal λ, so the standard deviation is √λ.
- λ
- Average number of events per interval (mean)
- k
- Number of events whose probability you want
- e
- Euler’s number ≈ 2.71828
- k!
- k factorial, 1 × 2 × … × k
Worked examples
Exactly 5 customers when the average is 3 per hour
P(X = 5) = 3⁵ × e⁻³ ÷ 5! = 243 × 0.049787 ÷ 120 ≈ 0.1008, about a 10% chance. Summing x = 0…5 gives P(X ≤ 5) ≈ 0.916, so there is roughly an 8.4% chance of more than five customers.
A help desk averaging 10 calls: 8 or fewer?
With λ = 10, the chance of exactly 8 calls is 10⁸ × e⁻¹⁰ ÷ 8! ≈ 0.1126. Adding x = 0…8 gives P(X ≤ 8) ≈ 0.333, so about two times in three the desk receives more than 8 calls.
No typos on a page (λ = 0.5)
P(X = 0) = e^(−0.5) ≈ 0.6065, so about 61% of pages are error-free and 39% contain at least one typo. P(X ≥ 0) is always 1.
Frequently asked questions
When should I use the Poisson distribution?+
Use it to model counts of independent events in a fixed interval of time or space when you know the average rate — arrivals, calls, defects, accidents or emails. Events must occur one at a time and not affect each other.
What is λ (lambda) in a Poisson distribution?+
λ is the average number of events in the interval you care about. It is both the mean and the variance of the distribution. Scale it to your interval: 12 per hour becomes 3 per 15 minutes.
What is the difference between P(X ≤ k) and P(X < k)?+
P(X ≤ k) includes exactly k events ("at most k"), while P(X < k) stops at k − 1 ("fewer than k"). Likewise P(X ≥ k) is "at least k" and P(X > k) is "more than k".
How is Poisson different from the binomial distribution?+
The binomial counts successes in a fixed number of trials n with probability p each. The Poisson has no fixed number of trials; it approximates the binomial well when n is large and p is small, with λ = n·p.
Can λ be a decimal?+
Yes. λ is an average, so it can be any positive number such as 0.5 or 3.7. The count k, however, must be a whole number because you cannot observe a fraction of an event.