The geometric distribution models the number of trials needed until the first success occurs in a sequence of independent Bernoulli trials (like repeated coin flips).

Think of it as the math of “How long until this finally works?”

🎯 When to Use It

Use the geometric distribution when:

• Each trial has two outcomes: success or failure
• The probability of success is constant each time
• Trials are independent
• You’re counting the trial number of the first success

Examples:

• How many coin flips until the first heads
• How many emails until someone replies
• How many customers until the first one buys something
• How many free‑throws until the first make

📌 Probability Formula

If is the probability of success on each trial, then:

Where:

• = trial number of the first success
• = 1, 2, 3, …

Interpretation:
You fail times, then succeed on the -th trial.

🧠 Expected Value and Variance

Meaning:
If success probability is 0.2, you expect the first success around trial .

⭐ Examples

Example 1: Coin Flip

Let success = “heads.”

Probability the first heads is on the 3rd flip:

Example 2: Basketball Free Throws

A player makes each free throw with probability .

Probability the first made shot is on attempt 4:

Expected number of attempts until the first make:

Example 3: Customer Purchases

A store estimates that each customer buys something with probability .

Probability the first purchase happens with the 6th customer:

Expected number of customers until the first purchase:

🎨 Intuition

The geometric distribution is like a video game where you keep trying a level until you finally beat it.
Each attempt is independent, and the chance of beating it stays the same.