Geometric Distribution Calculator

Geometric Distribution Formula:

\[ P(X=k) = (1-p)^{k-1} p \]

Probability P(X=k):

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1. What is 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:

\[ P(X=k) = (1-p)^{k-1} p \]

Where:

\( k \) — Number of trials until first success (positive integer)
\( p \) — Probability of success on each trial (0 ≤ p ≤ 1)

Explanation: The formula calculates the probability that the first success occurs on the k-th trial.

3. Importance of Geometric Distribution

Details: The geometric distribution is widely used in quality control, reliability engineering, and other fields where we want to know the probability of first success after a certain number of trials.

4. Using the Calculator

Tips: Enter the number of trials (k) as a positive integer and success probability (p) as a value between 0 and 1.

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 fixed number of trials.

Q2: What is the expected value for geometric distribution?
A: The expected number of trials until first success is E(X) = 1/p.

Q3: What's the variance of geometric distribution?
A: Variance is Var(X) = (1-p)/p².

Q4: What are real-world applications?
A: Used in modeling number of attempts needed to win a game, pass a test, or find a defective item.

Q5: How does it relate to exponential distribution?
A: Geometric is discrete counterpart to continuous exponential distribution.