Joint probability of two events calculator
See how likely two independent events are to happen together: probabilities of 0.6 and 0.25 multiply to 0.15. Enter each chance as a decimal between 0 and 1, so 0.35 rather than 35. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- P a
- 0.6
- P b
- 0.25
- Joint probability
- 0.15
The joint probability of two independent events is the chance that both events occur. Enter P(A) and P(B) as numbers from zero to one, and this calculator multiplies them to return P(A and B). It is designed for cases where independence is already known or explicitly assumed, such as separate random trials, unrelated component failures, or independently sampled outcomes. The calculation is deterministic, runs without network access, and rejects missing, non-finite, or out-of-range probabilities rather than producing a misleading result.
When multiplication gives the joint probability
For independent events, learning that one event occurred does not change the probability of the other. That relationship makes the multiplication rule valid: P(A and B) equals P(A) multiplied by P(B). If event A has probability 0.60 and event B has probability 0.25, their joint probability is 0.15.
In everyday terms, there is a fifteen percent chance that both occur when the events truly are independent. Independence is the essential condition, not a minor detail. Two events can each have valid marginal probabilities while still influencing one another, and multiplying those marginals in that situation can give the wrong answer.
For dependent events, the general rule uses a conditional probability: P(A and B) equals P(A) times P(B given A). Use this calculator only when the problem states that A and B are independent, or when a defensible model establishes independence.
How to enter probabilities and read the result
A value of zero means the event cannot occur under the model, one means it is certain, and values between them represent proportional likelihood. Use 0.35 for thirty-five percent, not 35. Boundary values behave naturally: if either input is zero, the joint is zero; if one input is one, the joint equals the other probability.
Inputs below zero or above one are rejected because they are not probabilities. Missing values, strings, infinity, and non-numeric values are also rejected instead of being silently converted. This strict input contract is useful in automated workflows because malformed percentages cannot quietly pass through as plausible results.
Checking assumptions and avoiding common mistakes
Before using the result, write down why the events are independent. Separate coin tosses under a stable process are the standard example, but real operational events often share causes. Two machines in the same building may both fail during one power outage; two purchases by the same customer may reflect the same preference; rain and traffic delays may be related.
In those cases, multiplying marginal probabilities understates or overstates the true joint chance. Another frequent mistake is confusing the probability that both events occur with the probability that at least one occurs. For independent events, the probability of both is P(A) times P(B), while the probability of at least one is one minus the product of their complementary probabilities.
They answer different questions. The returned JavaScript number follows standard floating-point arithmetic, so some decimal products may display tiny binary representation effects when consumed by other software. If a report needs a particular number of decimal places, apply its documented rounding policy after receiving the result rather than changing the probabilities before multiplication.
What you can do with it
Independent quality checks
Find the chance that two independently performed inspections both detect their respective defects.
Separate random trials
Calculate the probability that two independent games, draws, or experiments both produce specified outcomes.
Reliability modeling
Estimate the joint occurrence of two component events when the engineering model explicitly treats their failure mechanisms as independent.
FAQ
What formula does this calculator use?
For independent events it uses P(A and B) = P(A) × P(B).
Can I use percentages as inputs?
Convert percentages to decimals first. Enter 25 percent as 0.25, because each input must be from zero to one.
Does the calculator verify that the events are independent?
No. Independence cannot be determined from P(A) and P(B) alone; it must be established by the problem or model.
What if my events are not independent?
Then plain multiplication can understate or overstate the joint chance, as with two machines in the same building that fail in one power cut. Use the general rule instead: P(A) times the probability of B given that A happened.
What happens if one probability is 0 or 1?
If either probability is 0, the chance that both happen is 0. If one of them is 1, the joint result is simply the other probability, because the certain event adds no extra uncertainty.
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/joint-probability-two \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"p_a":0.6,"p_b":0.25}'const res = await fetch("https://api.kit.forhosting.com/stat/joint-probability-two", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"p_a": 0.6,
"p_b": 0.25
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/stat/joint-probability-two",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"p_a": 0.6,
"p_b": 0.25
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/stat/joint-probability-two", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"p_a":0.6,"p_b":0.25}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"p_a":0.6,"p_b":0.25}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/stat/joint-probability-two", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"p_a": 0.6,
"p_b": 0.25
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "stat.joint_probability_two",
"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
When multiplication gives the joint probability
The endpoint does not attempt to infer that relationship from the two inputs because two marginal probabilities alone contain no evidence about dependence. It performs the stated independent-events calculation exactly and reports the formula alongside the numeric result.
How to enter probabilities and read the result
Supply p_a and p_b as numeric probabilities on the closed interval from zero through one. The response echoes both validated inputs, returns joint_probability, and includes the multiplication formula for context. For example, p_a equal to 0.8 and p_b equal to 0.5 produce a joint probability of 0.4. Each successful API request costs $0.002, while the browser calculation is available locally without a paid request.
Checking assumptions and avoiding common mistakes
Also keep probabilities in decimal form throughout the request: convert 7 percent to 0.07 before sending it. That does not introduce randomness; repeated calls with the same inputs return the same JSON value.
Technical questions
What happens if a probability is outside zero to one?
The request fails with an invalid_input error that identifies the out-of-range field.
How much does an API calculation cost?
Each successful API request costs $0.002. The browser version runs locally for free.