Geometric Distribution Examples with Answers- Statistics Practice

What Is the Geometric Distribution?

The geometric distribution models the number of trials needed to get the first success in repeated independent trials. Each trial has two outcomes: success or failure, with constant probability p.

You see this in real situations like:

• How many times you roll a die before rolling a six
• How many job applications you send before landing an interview
• How many customers click through before someone makes a purchase

The distribution gives you the probability that your first success happens on trial number k.

The Geometric Distribution Formula

There are two equivalent formulations. Most textbooks use this one:

P(X = k) = (1 - p)^k-1 × p

Where:

• X = number of trials until first success
• k = specific trial count (k = 1, 2, 3...)
• p = probability of success on each trial
• (1 - p) = probability of failure

Mean and Variance

You need these for quick calculations:

• Mean (expected value) = 1/p
• Variance = (1 - p) / p²

Geometric Distribution Examples with Answers

Example 1: Rolling a Die

Problem: You roll a fair six-sided die. What is the probability that your first six appears on the 3rd roll?

Solution:

Here, p = 1/6 (probability of rolling a six).

Using P(X = k) = (1 - p)^k-1 × p

P(X = 3) = (5/6)² × (1/6)

P(X = 3) = 25/36 × 1/6

P(X = 3) = 25/216 ≈ 0.116 or 11.6%

Example 2: Manufacturing Defects

Problem: A factory produces widgets with a 4% defect rate. What is the probability that the first defective widget is the 10th one inspected?

Solution:

p = 0.04, k = 10

P(X = 10) = (0.96)⁹ × 0.04

P(X = 10) = 0.693 × 0.04

P(X = 10) ≈ 0.0277 or 2.77%

Example 3: Free Throw Shooting

Problem: A basketball player makes 75% of free throws. What is the expected number of attempts until she misses?

Solution:

Here "success" = miss, so p = 0.25

Expected value = 1/p = 1/0.25 = 4 attempts

On average, she'll make 3 free throws before missing one.

Example 4: Website Signups

Problem: A website converts 2% of visitors to subscribers. What is the probability that you get your first subscriber within the first 5 visitors?

Solution:

Calculate P(X ≤ 5) = 1 - P(X > 5) = 1 - (1 - p)⁵

P(X ≤ 5) = 1 - (0.98)⁵

P(X ≤ 5) = 1 - 0.904

P(X ≤ 5) ≈ 0.096 or 9.6%

Example 5: Variance Calculation

Problem: A coin is biased to land heads 30% of the time. Find the variance of the number of flips needed to get the first head.

Solution:

p = 0.30

Variance = (1 - p) / p² = 0.70 / 0.09

Variance ≈ 7.78

Geometric vs. Binomial vs. Negative Binomial

Students confuse these three constantly. Here's the difference:

The key distinction: geometric stops at one success. Negative binomial continues until r successes.

Common Mistakes Students Make

• Wrong formula version: Some textbooks use P(X = k) = (1-p)^kp for k = 0, 1, 2... Others use k = 1, 2, 3... Check which version your course uses.
• Confusing success and failure: Always identify what counts as "success" before plugging in numbers.
• Forgetting the exponent: P(X = 5) requires (1-p)⁴, not (1-p)⁵. The exponent is always k-1.
• Not checking independence: The geometric distribution requires independent trials. If previous outcomes affect future probabilities, you can't use it.

How to Solve Any Geometric Distribution Problem

Follow this step-by-step process:

Step 1: Identify the Parameters

Find p (probability of success) and k (number of trials). Write them down explicitly.

Step 2: Define Your Random Variable

State clearly what X represents. "X = number of trials until first success" is usually correct.

Step 3: Choose the Right Formula

For exact trial count: P(X = k) = (1-p)^k-1 × p

For "at most" questions: P(X ≤ k) = 1 - (1-p)^k

For "at least" questions: P(X ≥ k) = (1-p)^k-1

Step 4: Plug In and Calculate

Substitute your values. Use a calculator for powers. Round only your final answer.

Step 5: Check Your Work

All probabilities must be between 0 and 1. Verify that Σ P(X = k) = 1 for all possible values of k.

Practice Problems

Problem 1: A machine has a 15% failure rate per hour. What is the probability the first failure occurs within 2 hours?

Answer: P(X ≤ 2) = 1 - (0.85)² = 1 - 0.7225 = 0.2775 or 27.75%

Problem 2: A tennis player wins 60% of serves. How many serves should she expect to attempt before losing a point?

Answer: Expected value = 1/0.40 = 2.5 serves (she'll win about 1.5 serves on average before losing one)

Problem 3: An email campaign has a 5% open rate. What is the probability the first opened email is sent to the 4th recipient?

Answer: P(X = 4) = (0.95)³ × 0.05 = 0.857 × 0.05 = 0.0429 or 4.29%

When to Use Geometric Distribution

Use it when:

• Each trial is independent
• Each trial has only two outcomes
• The probability of success stays constant
• You're counting trials until the first success

Don't use it when outcomes are dependent, when you have a fixed number of trials, or when you need multiple successes.

That's the geometric distribution. The formula, examples, and common errors covered here should handle most homework and interview questions you'll encounter. 👍