Geometric Distribution Calculator

Calculate probabilities for the trial number of the first success in repeated Bernoulli trials

Calculate geometric distribution probabilities.

This tool uses the convention X = the trial number on which the first success occurs. It returns P(X = k), P(X <= k), P(X >= k), mean, variance, and standard deviation.

Example Results

1 examples

Find first success on the third trial

Calculate geometric probabilities when each trial succeeds with probability 20%.

{
  "result": {
    "exactProbability": 0.128,
    "atMostProbability": 0.488,
    "atLeastProbability": 0.64
  }
}
View input parameters
{ "successProbability": 20, "trialNumber": 3, "inputScale": "percent", "probabilityMode": "exact", "decimalPlaces": 4 }

Key Facts

Category
Math, Date & Finance
Input Types
number, select
Output Type
json
Sample Coverage
2
API Ready
Yes

Overview

The Geometric Distribution Calculator computes the probability of achieving the first success on a specific trial in a series of independent Bernoulli trials. By defining the success probability and the target trial number, you can instantly determine the exact, cumulative (at most), and survival (at least) probabilities, along with the distribution's mean, variance, and standard deviation.

When to Use

  • •Analyzing quality control processes to determine the likelihood of finding the first defective item on a specific inspection.
  • •Calculating the odds of achieving a successful outcome, such as a sale or a game win, on an exact attempt.
  • •Solving probability and statistics problems involving independent Bernoulli trials and first-success distributions.

How It Works

  • •Enter the probability of success for a single trial, choosing either a percentage or a decimal proportion.
  • •Input the target trial number (k) on which you want the first success to occur.
  • •Select the probability mode to focus on an exact trial, at most k trials, or at least k trials.
  • •Review the calculated probabilities, mean, variance, and standard deviation formatted to your preferred decimal places.

Use Cases

Quality Assurance: Estimating the probability that the first defective product is discovered exactly on the 10th item inspected.
Sales Forecasting: Calculating the likelihood of closing the first deal on the 5th cold call given a known conversion rate.
Gaming & Loot Drops: Determining the chances of obtaining a rare item within the first 3 attempts when the drop rate is fixed.

Examples

1. Quality Control Inspection

QA Engineer
Background
A manufacturing line has a known defect rate of 5%. The QA engineer wants to know the probability that the first defective item is found exactly on the 4th inspection.
Problem
Calculate the exact probability of the first success (finding a defect) on trial 4.
How to Use
Set Success Probability to 5, Trial Number to 4, Input Scale to Percent, and Probability Mode to Exactly on Trial k.
Example Config
Success Probability: 5, Trial Number: 4, Input Scale: percent, Probability Mode: exact
Outcome
The calculator returns an exact probability of 0.0429 (4.29%), showing the likelihood of the first defect appearing on the fourth check.

2. Sales Conversion Odds

Sales Representative
Background
A sales rep has a 20% success rate for cold calls. They want to know the probability of making their first sale within the first 3 calls.
Problem
Find the cumulative probability of getting the first success in 3 or fewer trials.
How to Use
Enter 20 for Success Probability, 3 for Trial Number, select Percent for Input Scale, and choose 'At Most k Trials' for Probability Mode.
Example Config
Success Probability: 20, Trial Number: 3, Input Scale: percent, Probability Mode: at-most
Outcome
The result shows an 'at most' probability of 0.488 (48.8%), meaning there is nearly a 50% chance of closing a deal within the first three calls.

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FAQ

What is a geometric distribution?

It is a discrete probability distribution that models the number of independent trials needed to get the first success, where each trial has the same probability of success.

What does 'trial number' mean in this calculator?

It represents the total number of attempts made up to and including the first successful outcome. This calculator uses the convention where X equals the trial number of the first success.

Can I input the success probability as a percentage?

Yes, you can use the 'Input Scale' option to enter the success probability as either a percentage (e.g., 20%) or a decimal proportion (e.g., 0.20).

What is the difference between 'At Most' and 'At Least' modes?

'At Most k Trials' calculates the cumulative probability that the first success happens on or before trial k. 'At Least k Trials' calculates the probability that it takes k or more trials to succeed.

Are the mean and variance calculated automatically?

Yes, the calculator automatically computes the expected value (mean), variance, and standard deviation based on the provided success probability.

API Documentation

Request Endpoint

POST /en/api/tools/geometric-distribution-calculator

Request Parameters

Parameter Name Type Required Description
successProbability number Yes -
trialNumber number Yes -
inputScale select No -
probabilityMode select No -
decimalPlaces number No -

Response Format

{
  "key": {...},
  "metadata": {
    "key": "value"
  },
  "error": "Error message (optional)",
  "message": "Notification message (optional)"
}
JSON Data: JSON Data

AI MCP Documentation

Add this tool to your MCP server configuration: 

{
  "mcpServers": {
    "elysiatools-geometric-distribution-calculator": {
      "name": "geometric-distribution-calculator",
      "description": "Calculate probabilities for the trial number of the first success in repeated Bernoulli trials",
      "baseUrl": "https://elysiatools.com/mcp/sse?toolId=geometric-distribution-calculator",
      "command": "",
      "args": [],
      "env": {},
      "isActive": true,
      "type": "sse"
    }
  }
}

You can chain multiple tools, e.g.: `https://elysiatools.com/mcp/sse?toolId=png-to-webp,jpg-to-webp,gif-to-webp`, max 20 tools.

If you encounter any issues, please contact us at [email protected]