Poisson at least probability calculator
With an average of 2.5 events per interval, the chance of at least 3 is 0.4561868841166705. Enter the average rate and the minimum count you care about to get the upper-tail Poisson probability for tickets, failures or claims. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Lambda
- 2.5
- K
- 3
- Probability
- 0.4561868841
The Poisson distribution models how many independent events land in a fixed interval when the average rate stays constant. Call centers, reliability teams, and product analysts often need more than the chance of exactly k arrivals: they need the chance of at least k tickets, failures, or claims. This Poisson at least probability calculator takes a non-negative rate lambda and a non-negative integer k, then returns the upper-tail probability P(X is greater than or equal to k). Negative rates and negative counts are rejected with a clear validation error so queueing models never emit silent nonsense.
What P(X ≥ k) means under a Poisson rate
A Poisson random variable X with rate parameter lambda takes the non-negative integers 0, 1, 2, and so on. The familiar mass function gives the chance of exactly i events as e to the minus lambda times lambda to the power i over i factorial. Summing that mass from zero through m yields the left cumulative probability of at most m events.
The upper-tail question that staffing, SLA, and rare-event thresholds usually ask is the complement: what is the probability of at least k events in the same interval? That is P(X greater than or equal to k). For k equal to zero the answer is always one, because every non-negative count is at least zero.
For every positive integer k the identity is P(X greater than or equal to k) equals one minus P(X less than or equal to k minus one), which reuses a stable left cumulative sum without enumerating an infinite right tail. When lambda is zero the mass sits entirely at zero, so the at-least probability is one for k equal to zero and zero for every positive k.
When lambda is positive the mass peaks near the rate and the upper tail thins as k moves far above lambda, which is exactly the language operations teams use when they ask how often demand can spike past a staffing floor. Because the formulas are classical and closed-form in the discrete sum sense, the same lambda and k always produce the same probability, which makes the result safe to pin in homework keys, automated tests, and reproducible notebooks without Monte Carlo noise.
How this calculator evaluates the Poisson upper tail
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 minimum count of interest. After validation, when k is zero the probability is returned as one immediately. Otherwise the core builds the left cumulative P(X less than or equal to k minus one) by starting from e to the minus lambda and multiplying successive terms by lambda over i, then subtracts that cumulative from one and clamps the result into the unit interval.
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 complement still settles.
Practical use, limits, pricing, and delivery
Use the browser tool when you want an immediate answer to “what is the chance of at least k arrivals, defects, or claims?” while checking a queueing homework item, 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.
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.
What you can do with it
Staffing and SLA upper-tail thresholds
Operations teams convert a measured arrival rate into P(X ≥ k) to set coverage floors and service-level breach odds without simulation.
Homework and statistics labs on Poisson tails
Students and graders verify the complement identity P(X ≥ k) = 1 − P(X ≤ k−1) for given lambda and non-negative integer k.
Reliability and rare-event monitoring
Engineers obtain stable defect or failure upper-tail probabilities for alert 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.
How is P(X ≥ k) related to the left cumulative Poisson probability?
For k ≥ 1, P(X ≥ k) equals 1 − P(X ≤ k − 1).
Is the Poisson at least probability calculator free?
The browser tool runs free on your device.
Does this simulate random Poisson samples?
No. It evaluates the closed-form upper-tail probability deterministically. The same lambda and k always return the same probability.
How do I find the chance that demand exceeds my staffing level?
Enter the average number of arrivals per interval as the rate and, as k, the smallest count your team cannot absorb. The result is the share of intervals in which at least that many tickets, failures or claims arrive.
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/stat/poisson-at-least \
-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/stat/poisson-at-least", {
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/stat/poisson-at-least",
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/stat/poisson-at-least", 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/stat/poisson-at-least", 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": "stat.poisson_at_least",
"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.
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.
How this calculator evaluates the Poisson upper tail
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. The response echoes lambda and k, returns the upper-tail probability, and includes a short formula string restating the complement identity.
Practical use, limits, pricing, and delivery
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 upper-tail probability twice.
Technical questions
How do I compute the Poisson probability of at least k events?
Send JSON with non-negative number lambda (rate) and non-negative integer k to stat.poisson_at_least. The response returns probability P(X ≥ k) and a short formula string.