# Geometric Distribution Calculator

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

> Canonical page: https://elysiatools.com/en/tools/geometric-distribution-calculator

- **Category:** Math & Numbers

- **Keywords:** geometric distribution, first success, bernoulli trials, probability, statistics

## Overview

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.

## Inputs

- **Success Probability** (number)
- **Trial Number** (number)
- **Input Scale** (select)
- **Probability Mode** (select)
- **Decimal Places** (number)

## 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.

## Frequently asked questions

### 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.

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