1. What is the Geometric Distribution?

The geometric distribution describes the probability distribution of the number of trials needed to get the first success in repeated, independent Bernoulli trials with success probability p.

2. How Does the Calculator Work?

The calculator uses the geometric distribution formula:

Where:

• \( p \) — Probability of success on an individual trial (0 < p ≤ 1)
• \( x \) — Trial number of first success (x ≥ 1)

Explanation: The probability decreases exponentially with each additional trial needed to get the first success.

3. Importance of Geometric Distribution

Details: The geometric distribution is used in quality control, reliability engineering, and other fields where we want to know how many trials are needed before the first success occurs.

4. Using the Calculator

Tips: Enter the probability of success (must be between 0 and 1) and the trial number (must be ≥1). The calculator will compute the probability that the first success occurs on trial x.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between geometric and binomial distributions?
A: Geometric counts trials until first success, while binomial counts successes in a fixed number of trials.

Q2: What is the expected value of a geometric distribution?
A: The mean is 1/p, meaning on average you need 1/p trials to get the first success.

Q3: What are real-world applications of geometric distribution?
A: Modeling number of sales calls needed to make first sale, number of parts tested until first defective, etc.

Q4: What happens when p=1?
A: The distribution becomes deterministic - you always succeed on the first trial.

Q5: How does the distribution change with different p values?
A: Higher p makes early successes more likely (distribution decays faster), lower p makes later successes more probable.