Geometric Distribution Calculator

Geometric Distribution Calculator | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Math & Numbers
Input types: number, select
Output type: json
Sample coverage: 4
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

1Enter the probability of success for a single trial, choosing either a percentage or a decimal proportion.
2Input the target trial number (k) on which you want the first success to occur.
3Select the probability mode to focus on an exact trial, at most k trials, or at least k trials.
4Review 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.

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.

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?

Success Probability: 20, Trial Number: 3, Input Scale: percent, Probability Mode: at-most

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Quality Control Inspection

2. Sales Conversion Odds

FAQ

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