Poisson probability calculator
If tickets arrive at an average of 2.5 per hour, exactly 3 in an hour happens a little over one time in five, and 3 or fewer about three times in four. Enter the rate and the count to get both, plus mean and variance. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Poisson rate parameter λ
- 2.5
- K
- 3
- Probability
- 0.2137630172
- Cumulative
- 0.7575761331
- Mean
- 2.5
- Variance
- 2.5
The Poisson distribution models how many discrete events occur in a fixed interval when arrivals are independent and the average rate is constant. Call centers track tickets per hour, reliability engineers count failures per month, and analysts measure clicks, defects, or rare biological events with the same two parameters: a non-negative rate lambda and an integer count k. This calculator returns the exact Poisson probability of observing k events, the cumulative probability of at most k events, and the theoretical mean and variance (both equal to lambda). Negative rates and negative counts are rejected with a clear validation error so malformed queueing or counting models never produce silent nonsense numbers.
What the Poisson probability mass and cumulative functions mean
A Poisson random variable X with rate parameter lambda takes non-negative integer values 0, 1, 2, and so on. The probability of observing exactly k events in the reference interval is given by the mass function P(X equals k) equals e to the power of minus lambda, times lambda to the power k, divided by k factorial. When lambda is zero the entire mass sits at zero, so P(X equals zero) is one and every positive k has probability zero.
The cumulative distribution function F of k is the sum of the mass from zero through k inclusive, so it answers the everyday question “what is the chance of at most k arrivals, defects, or claims?” Because the mean and the variance of a Poisson law both equal lambda, a single rate parameter completely determines location and spread. That property is why introductory statistics courses introduce Poisson models right after binomial approximations, and why operations teams can quote service-level odds from a measured average alone.
The formulas are classical and fully deterministic: the same lambda and k must always produce the same probability, the same cumulative value, and the same mean and variance, which makes the numbers safe to pin in homework keys, automated tests, and reproducible notebooks without Monte Carlo noise.
How this calculator evaluates P(X=k), the CDF, mean, and variance
Call the capability with two required fields: lambda, a finite non-negative number for the average event rate, and k, a non-negative integer for the event count of interest. After validation, the core evaluates the probability mass in log space as minus lambda plus k times the natural log of lambda minus the log of k factorial, then exponentiates to a stable floating probability.
The cumulative value is built by starting from e to the minus lambda and multiplying successive terms by lambda over i, which avoids recomputing large factorials from scratch for every index. When e to the minus lambda underflows for very large rates, the engine falls back to a log-space sum of individual mass terms so the cumulative still settles. Mean and variance are reported as lambda itself, matching the closed Poisson identities.
What you can do with it
Queueing and operations staffing
Support and logistics teams convert a measured arrival rate into P(X=k) and P(X≤k) to set shift coverage and service-level targets without simulation.
Homework and introductory statistics labs
Students and graders verify Poisson mass and cumulative values for given lambda and k, including the identity that mean and variance both equal the rate.
Reliability and rare-event checks
Engineers and product analysts obtain stable defect, failure, or click-count probabilities for monitoring thresholds and automated test oracles.
FAQ
What happens if lambda or k is negative?
Negative rates and negative event counts are rejected and not charged; only lambda ≥ 0 and integer k ≥ 0 are accepted.
What are the mean and variance of a Poisson distribution?
Both equal the rate parameter lambda. This calculator reports mean and variance as lambda after validating the inputs.
Is the Poisson probability calculator free?
The browser tool runs free on your device.
Does this simulate random Poisson samples?
No. It evaluates the closed-form mass and cumulative functions deterministically. The same lambda and k always return the same probability and cumulative value.
How do I get the chance of at most k events?
That is the cumulative value the calculator returns: the sum of the probabilities from zero through k. With a rate of 2.5 and at most 3 events it is about three in four, useful for quoting service levels from a measured average.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/prob/poisson-distribution \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"lambda":2.5,"k":3}'const res = await fetch("https://api.kit.forhosting.com/prob/poisson-distribution", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"lambda": 2.5,
"k": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/prob/poisson-distribution",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"lambda": 2.5,
"k": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/prob/poisson-distribution", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"lambda":2.5,"k":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"lambda":2.5,"k":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/prob/poisson-distribution", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"lambda": 2.5,
"k": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "prob.poisson_distribution",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_lambda | 1000 |
max_k | 10000 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
The arithmetic is fully deterministic, runs free in your browser, and is available over the API at $0.002 per successful request.
What the Poisson probability mass and cumulative functions mean
When lambda is positive the distribution is unimodal and concentrated near the rate: for small lambda the chance of zero or one event dominates, while larger rates push probability mass toward higher counts and flatten the peak.
How this calculator evaluates P(X=k), the CDF, mean, and variance
The engine also accepts the aliases rate and mean for the same rate field so scripts that already speak in those terms do not need a rename step. Validation rejects missing fields, non-finite values, non-integer k, lambda below zero, k below zero, and values above the published maxima, all with invalid_input and without charging balance.
Practical use, limits, pricing, and delivery
Use the browser tool when you want an immediate Poisson probability while checking a queueing homework answer, sizing a support shift, or explaining rare-event odds without installing a statistics package. Results are fully deterministic: the same lambda and k always yield the same probability, cumulative, mean, and variance. There is no language model, no external probability service, and no randomness, so the capability is safe to pin in continuous-integration checks and reproducible lesson materials. Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted ranges keep the recursive sum bounded for Worker isolates: rates up to one thousand and counts up to ten thousand cover common operational and classroom examples while still finishing quickly. Wire the endpoint into capacity-planning tools, defect-rate tutorials, coding-interview sandboxes, or event-count pipelines where you already prefer closed-form Poisson arithmetic over approximate simulation histograms that never quite settle on the same tail probability twice.
Technical questions
How do I compute a Poisson probability with this calculator?
Send JSON with non-negative number lambda (rate) and non-negative integer k to prob.poisson_distribution. The response returns probability P(X=k), cumulative P(X≤k), mean, and variance.