About this calculator
Geometric Distribution Calculator is a professional probability and statistics tool used to calculate the probability, expectation and variance of geometric distributions. The geometric distribution describes the probability distribution of the number of trials required for the first success in a Bernoulli trial. For example, flipping a coin until the first heads comes up, or drawing a lottery until the first win occurs. Geometric distribution is a discrete probability distribution that is widely used in fields such as reliability analysis, quality control, and queuing theory. This calculator can calculate the probability, cumulative probability, expected value, variance and other statistics of a specific number of times, and provide probability distribution charts.
What it calculates
The geometric distribution calculator finds the probability that the first success occurs on trial k.
Formula
P(X = k) = (1-p)^(k-1) p, where p is the success probability for one trial.
Inputs
- Success probability p.
- Trial number k of the first success.
Example
| p | k | Probability expression |
|---|---|---|
| 0.5 | 3 | 0.5^2*0.5 |
| 0.2 | 1 | 0.2 |
| 0.1 | 5 | 0.9^4*0.1 |
How to interpret the result
The probability means the first k-1 trials fail and the kth trial succeeds. As k grows, the probability often gets smaller.
Common mistakes
- k starts at 1, not 0.
- Trials should be independent with fixed p.
- Do not confuse it with binomial probability for a fixed number of successes.
How to use
Use the geometric distribution calculator:
1. Enter the success probability p (0<p≤1) 2. Select the calculation type: • P(X=k): Probability of exactly kth success • P(X≤k): The cumulative probability of no more than k successes • P(X>k): The probability of success after more than k times 3. Enter the number of tests k 4. Click the "Calculate" button 5. View the results: • Probability value • Expect E(X)=1/p • Variance Var(X)=(1-p)/p² • Probability distribution plot
Main features
• Multiple probabilities: calculate point and cumulative probabilities • Statistics: automatic calculation of expectation and variance • Distribution plot: Visualize probability distributions • Formula display: display calculation formulas • Parameter validation: Check input validity • Example description: Provide application examples • Comparative analysis: compared with other distributions • Totally free: unlimited use
Use cases
• Reliability analysis: Calculate time to first failure • Quality control: analysis of first-time non-conforming products • Lottery problem: Calculate the probability of winning for the first time • Queuing Theory: Analyzing Waiting Times • Market Research: First Time Buying Behavior • Experimental design: planning the number of experiments • Probability teaching: explaining geometric distribution • Data analysis: fitting geometric distributions