Geometric Distribution Calculator - Probability Analysis
Calculate geometric distribution probabilities for trials until first success, including exact and cumulative probabilities, mean, and variance
Distribution Parameters
Results
What is a Geometric Distribution Calculator?
A Geometric Distribution Calculator computes probabilities for the number of trials needed to achieve the first success in a series of independent Bernoulli trials.
This calculator is used for:
- Quality Control - Trials until first defective item
- Sales Analysis - Number of calls until first sale
- Reliability Testing - Time until first failure
- Game Theory - Attempts until winning
To calculate probabilities for fixed numbers of trials, explore our Binomial Distribution Calculator to determine probabilities for multiple successes in repeated experiments.
To work with continuous time-based events, check out our Exponential Distribution Calculator to model waiting times and reliability in continuous processes.
To apply continuity corrections for discrete distributions, visit our Continuity Correction Calculator to improve normal approximations for discrete probabilities.
To estimate population parameters with confidence, use our Confidence Interval Calculator to determine statistical ranges for means and proportions.
How Geometric Distribution Works
The geometric distribution uses these formulas:
Where:
- k = Trial number of first success
- p = Probability of success (0 < p ≤ 1)
- (1 - p) = Probability of failure
Key Concepts Explained
First Success
Geometric distribution models the waiting time until the first success. Each trial is independent with constant success probability.
Memoryless Property
Past failures don't affect future probabilities. If first k trials fail, probability distribution resets for remaining trials.
Expected Trials
Mean = 1/p gives expected number of trials until first success. If p = 0.25, expect 4 trials on average.
How to Use This Calculator
Enter Success Probability (p)
Input probability of success on each trial (0 to 1)
Enter Number of Trials (k)
Input the trial number to analyze
View Probabilities
Review exact, cumulative probabilities, mean, and variance
Benefits of This Calculator
- Complete Analysis - Exact and cumulative probabilities
- Statistical Measures - Mean, variance, standard deviation
- Instant Results - Immediate probability calculations
- Accurate Formulas - Uses standard geometric formulas
- Quality Control - Perfect for reliability analysis
- Educational Tool - Great for probability students
Important Considerations
- Success Probability - p must be between 0 and 1 (exclusive of 0)
- Independence - Each trial must be independent
- Constant Probability - p remains same for all trials
- Binary Outcomes - Only two outcomes: success or failure
- Low Probability - Small p means high expected trials (1/p)
- High Variance - Lower p creates more variable results
Frequently Asked Questions
What is a geometric distribution?
The geometric distribution models the number of trials needed to get the first success in a series of independent Bernoulli trials, each with the same probability of success p.
What is the formula for geometric distribution?
The probability mass function is P(X = k) = (1 - p)^(k-1) × p, where k is the number of trials needed for the first success and p is the probability of success on each trial.
What is the mean of a geometric distribution?
The mean (expected value) of a geometric distribution is μ = 1/p, where p is the probability of success. This represents the expected number of trials until the first success.
What is the variance of a geometric distribution?
The variance of a geometric distribution is σ² = (1 - p) / p², where p is the probability of success. The standard deviation is σ = √((1 - p) / p²).
When is geometric distribution used?
Geometric distribution is used for modeling the number of attempts until first success, such as: number of products tested until finding a defect, sales calls until first sale, coin flips until first heads, or trials until first occurrence of an event.
What is the memoryless property of geometric distribution?
The geometric distribution has the memoryless property: if the first success hasn't occurred after k trials, the probability distribution for future trials is the same as if starting fresh. Past failures don't affect future probabilities.