SigmaCalc

Poisson Distribution Calculator

The distribution behind call-center volume, website traffic spikes, and any 'how many events in this time window' question.

Average rate (λ)

Number of events (k)

P(X = k)

0.195

P(X = k) = e⁻λ × λᵏ ÷ k!

What you entered

Average rate (λ): 4Number of events (k): 3

1
Apply the Poisson formula
P(X = 3) = e^(−4) × 4^3 ÷ 3!= 0.1954

Result

P(X = k): 0.195

If events happen at an average rate of 4 per interval, there's a 19.54% chance of seeing exactly 3 in a given interval.

Poisson probability of exactly k events

The Poisson distribution models counts of events in a fixed interval when they occur independently at a constant average rate lambda.

Lambda (mean rate)	Events k	Probability
2	0	0.1353
2	1	0.2707
2	2	0.2707
3	3	0.2240
4	2	0.1465
5	5	0.1755

Rows two and three are identical at 0.2707, which is not a coincidence: for a Poisson distribution, P(k) and P(k-1) are equal exactly when k equals lambda, so with a mean of 2 the probabilities of 1 and 2 events always match. Notice too that even at the most likely count the probability is modest - with a mean of 5, exactly 5 events happens only 17.55% of the time. Poisson fits arrivals at a queue, defects per batch, or emails per hour, provided events are independent and the rate is stable. If events cluster or the rate varies through the interval, it will understate the extremes.

When Poisson fits

It models the count of independent, rare-ish events over a fixed interval of time or space, where events happen at a known average rate but the exact timing is random — customer arrivals, typos per page, server errors per hour.

Rate and interval must match

λ (lambda) is the average number of events in the same interval you're asking about — if your rate is per hour but you want a probability for a 10-minute window, convert λ to match that window first.

Frequently asked questions

A call center gets 4 calls per minute on average. What's the probability of getting 7 calls in one minute?

λ=4, k=7. P(X=7) = e⁻⁴ × 4⁷ / 7! = 0.0183 × 16384 / 5040 = 0.0595 = 5.95%. Getting 7+ calls is uncommon but not rare — important for staffing decisions.

How is Poisson different from binomial?

Binomial counts successes in a fixed number of trials (5 heads in 10 flips). Poisson counts events in a fixed time/space interval (3 customers in an hour). Use Poisson when events happen continuously at a rate, not in discrete yes/no trials.

A website averages 2 server errors per day. What's the chance of a zero-error day?

λ=2, k=0. P(X=0) = e⁻² × 2⁰ / 0! = 0.1353 = 13.5%. About 1 in 7 days will be error-free. P(3+ errors) = 1 − P(0) − P(1) − P(2) = 1 − 0.135 − 0.271 − 0.271 = 32.3%.