At least one event probability calculator
The at least one event probability calculator finds the chance that an event happens one or more times when every trial is independent and has the same probability of success.
Run — free
Enter the probability for a single trial as a decimal between zero and one, then enter the number of trials. The result uses the complement rule: it calculates the chance of no successes and subtracts that value from one. This is often simpler and more reliable than adding the probabilities for one success, two successes, and every other possible successful count.
Understand the complement method
Finding the probability of at least one success directly can become tedious because the phrase includes many possible outcomes. With five trials, for example, it covers exactly one success, exactly two, exactly three, exactly four, and exactly five. The complement contains only one outcome: no successes at all. If the success probability on one trial is p, the failure probability is 1 - p. For n independent trials, the probability that every trial fails is (1 - p)^n. Subtracting that value from one gives 1 - (1 - p)^n, the probability that one or more trials succeed. The calculator returns both quantities so you can see the relationship and check that they add to one, apart from normal floating-point precision at extreme values. A probability of zero always produces zero for at least one success. A probability of one produces certainty whenever there is at least one trial. Zero trials produce zero successes by definition, so their at-least-one probability is zero regardless of the per-trial probability.
Enter probability and trials correctly
Use a decimal probability between 0 and 1 inclusive. For example, enter 0.25 for a twenty-five percent chance on each trial; do not enter 25, because that lies outside the accepted probability range. The trials value must be a non-negative whole number and may be zero. Independence is essential: the outcome of one trial must not change the probability on another trial, and the same per-trial probability must apply throughout. Repeated fair die rolls and repeated draws with replacement are common examples. Drawing cards without replacement is not independent because the deck changes after every draw, so this formula would not model that experiment correctly. Likewise, repeated attempts made under changing weather, inventory, fatigue, or system load may have different probabilities. In those cases, use a model that accepts a separate probability for each attempt. The calculator validates the numerical range but cannot determine from the numbers alone whether your real-world trials are independent, so that assumption remains part of your interpretation.
Interpret and use the result
The primary result is a decimal probability from 0 to 1. Multiply it by 100 if you want a percentage for a report or dashboard. The returned probability of no success is useful as a cross-check and may also be the operational risk you care about, such as the chance that every redundant component fails to respond. More trials increase the probability of at least one success when the per-trial probability is greater than zero, but they do not guarantee that every trial succeeds or describe how many successes to expect. The formula answers only whether the count is at least one. For planning, compare candidate trial counts while keeping the single-trial probability fixed, or compare interventions that improve the per-trial probability while keeping the count fixed. Avoid rounding the input before calculating when the event is rare or the trial count is large, because a small change in p can materially change the combined probability. The implementation uses logarithmic complement arithmetic to preserve useful precision for very small probabilities and many trials. Browser use is free; API automation costs $0.002 per request.
What you can do with it
Estimate detection coverage
Find the chance that at least one independent sensor or repeated inspection detects an event when each attempt has the same detection probability.
Plan repeated outreach
Estimate the chance of at least one response across independent attempts, provided the response probability stays constant and attempts do not influence one another.
Evaluate reliability
Calculate the probability that at least one independent redundant component succeeds when each component has the same success probability.
FAQ
What formula does the calculator use?
It uses 1 - (1 - p)^n, where p is the success probability for one trial and n is the number of independent trials.
Can I enter a percentage such as 20?
No. Enter probabilities as decimals from 0 through 1, so twenty percent is 0.2. Values outside that range return an input error.
Why must the trials be independent?
Independence allows the no-success probabilities to be multiplied. If one outcome changes later probabilities, this formula does not describe the experiment.
What happens when the number of trials is zero?
With no trials there can be no success, so the probability of at least one success is 0 and the probability of no success is 1.
Does the result give the expected number of successes?
No. It gives the probability that the success count is one or greater. The expected count for this model is n multiplied by p.
What does API use cost?
Each API request costs $0.002. The same deterministic calculation is available free in the browser.
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/at-least-one \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"probability":0.2,"trials":5}'const res = await fetch("https://api.kit.forhosting.com/stat/at-least-one", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"probability": 0.2,
"trials": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/stat/at-least-one",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"probability": 0.2,
"trials": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/stat/at-least-one", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"probability":0.2,"trials":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"probability":0.2,"trials":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/stat/at-least-one", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"probability": 0.2,
"trials": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "stat.at_least_one",
"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_trials | 1000000000 |
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. |