Geometric Distribution

Distributions and Statistical Models

Calculate probabilities related to the number of trials required for the first success in a series of Bernoulli trials.

Examples

Explore some real-world scenarios to understand how the geometric distribution works.

First Successful Shot

Basketball Free Throws

A basketball player has a 75% chance of making a free throw. What is the probability that they make their first successful shot on their 3rd attempt?

p: 0.75, k: 3

Type: P(X = k) - Probability of first success on the k-th trial.

Finding a Defective Product

Quality Control

The probability of a manufactured item being defective is 5%. What is the probability that the first defective item is found within the first 10 items inspected?

p: 0.05, k: 10

Type: P(X ≤ k) - Probability of first success on or before the k-th trial.

Rolling a Six

Rolling a Die

You are rolling a fair six-sided die. What is the probability that you need more than 4 rolls to get your first six?

p: 0.1667, k: 4

Type: P(X > k) - Probability of first success after the k-th trial.

First Click-Through

Email Marketing

An email campaign has a 20% click-through rate. What's the probability the first click occurs on the 5th email sent or later?

p: 0.20, k: 5

Type: P(X ≥ k) - Probability of first success on or after the k-th trial.

Other Titles
Understanding the Geometric Distribution: A Comprehensive Guide
Dive deep into the principles, applications, and calculations of the geometric distribution, a fundamental concept in probability and statistics.

What is the Geometric Distribution?

  • Defining the Core Concept
  • Key Characteristics and Assumptions
  • Two Variations of the Distribution
The geometric distribution is a discrete probability distribution that models the number of successive, independent Bernoulli trials needed to achieve the first success. A Bernoulli trial is a random experiment with exactly two possible outcomes: 'success' or 'failure', where the probability of success is the same for every trial. This distribution is fundamental for analyzing 'waiting time' problems until a specific event occurs.
Key Characteristics and Assumptions
For a random variable to follow a geometric distribution, it must satisfy four key conditions: the trials must be independent, each trial must have only two outcomes (success or failure), the probability of success (p) must be constant for each trial, and the variable of interest is the number of trials required to get the first success.
Two Variations of the Distribution
It's important to distinguish between two common forms of the geometric distribution. One version models the number of trials (k) needed to get the first success (k = 1, 2, 3, ...). The other models the number of failures (y = k - 1) before the first success (y = 0, 1, 2, ...). Our calculator focuses on the first version, which is more common in introductory statistics.

Conceptual Examples

  • Flipping a coin until you get the first head.
  • Rolling a die until you land a 6.
  • A salesperson making calls until they make their first sale.

Step-by-Step Guide to Using the Geometric Distribution Calculator

  • Inputting Your Data Correctly
  • Choosing the Right Calculation Type
  • Interpreting the Results
Our calculator simplifies complex formulas into an easy-to-use interface. Follow these steps to get accurate results.
Inputting Your Data Correctly
Start by entering the 'Probability of Success (p)', which must be a number between 0 and 1. Then, enter the 'Number of Trials (k)', which represents the trial on which you're interested in the first success occurring. This must be a positive integer.
Choosing the Right Calculation Type
Select one of the four probability types: P(X = k) for the exact trial, P(X ≤ k) for success on or before a trial, P(X > k) for success after a trial, or P(X ≥ k) for success on or after a trial. Your choice depends on the specific question you're trying to answer.
Interpreting the Results
The calculator provides the main calculated probability, along with key statistical measures like the mean (expected number of trials), variance, and standard deviation. The distribution table offers a broader view by showing the probability of the first success occurring on different trials.

Example Calculation Walkthrough

  • If p=0.2 and k=3 for P(X=k), the calculator finds the probability of the first success being exactly on the third trial.
  • If p=0.1 and k=5 for P(X≤k), it calculates the sum of probabilities for the first success occurring on trial 1, 2, 3, 4, or 5.

Real-World Applications of the Geometric Distribution

  • Business and Quality Control
  • Science and Research
  • Everyday Life
The geometric distribution is not just a theoretical concept; it has numerous practical applications across various fields.
Business and Quality Control
In manufacturing, it can be used to determine the expected number of items to inspect before finding a defective one. This helps in planning quality assurance processes and resource allocation.
Science and Research
In biology, it can model the number of attempts needed for a certain experimental outcome, like successful gene replication. In medicine, it could model the number of treatments required before a patient responds positively.
Everyday Life
The distribution appears in many everyday scenarios, such as the number of times you have to roll a die to get a specific number, or the number of job applications you need to submit before getting an offer.

Application Scenarios

  • A market researcher asking people a survey question until they find someone who agrees with a certain viewpoint.
  • An angler casting a line until they catch their first fish.

Mathematical Derivation and Formulas

  • The Probability Mass Function (PMF)
  • The Cumulative Distribution Function (CDF)
  • Mean, Variance, and Standard Deviation
Understanding the formulas behind the geometric distribution provides deeper insight into how the probabilities are calculated.
The Probability Mass Function (PMF)
The probability of the first success occurring on the k-th trial is given by the formula: P(X = k) = (1 - p)^(k-1) * p. This represents the probability of (k-1) failures followed by one success.
The Cumulative Distribution Function (CDF)
The probability of the first success occurring on or before the k-th trial is given by: P(X ≤ k) = 1 - (1 - p)^k. This is often easier to calculate than summing individual PMF values.
Mean, Variance, and Standard Deviation
The key statistical measures are calculated as follows: Mean (μ) = 1/p. Variance (σ²) = (1 - p) / p². The Standard Deviation (σ) is the square root of the variance.

Formula Examples

  • For p=0.25, the mean is 1/0.25 = 4. You'd expect to wait 4 trials for the first success.
  • For p=0.5, the variance is (1-0.5)/0.5² = 0.5/0.25 = 2.