Geometric Distribution Calculator
Calculate geometric distribution probabilities and statistics
Table of Contents
How to Use
- Enter the probability of success for each trial (between 0 and 1)
- Enter the trial number you want to analyze
- Click calculate to see probabilities and statistics
- Review the distribution table for multiple trials
What is Geometric Distribution?
The geometric distribution models the number of independent trials needed to achieve the first success in a sequence of Bernoulli trials. Each trial has the same probability of success p, and trials are independent.
For example, if you're flipping a coin until you get heads, or rolling a die until you get a six, you're dealing with a geometric distribution.
Key Formulas
| Measure | Formula | Description |
|---|---|---|
| Probability Mass | P(X = k) = (1-p)^(k-1) × p | Probability of first success on trial k |
| Cumulative (≤) | P(X ≤ k) = 1 - (1-p)^k | Probability of success within k trials |
| Cumulative (≥) | P(X ≥ k) = (1-p)^(k-1) | Probability of needing k or more trials |
| Mean | μ = 1/p | Expected number of trials until first success |
| Variance | σ² = (1-p)/p² | Measure of spread |
| Standard Deviation | σ = √[(1-p)/p²] | Square root of variance |
Properties of Geometric Distribution
- Memoryless property: Past failures don't affect future probabilities
- Only defined for positive integers (k = 1, 2, 3, ...)
- Probability decreases exponentially as k increases
- Mean is always greater than or equal to 1
- Higher success probability p leads to lower expected trials
Real-World Applications
- Quality control: Testing items until finding a defect
- Customer service: Calls until reaching a representative
- Sales: Contacts until making a sale
- Medical trials: Treatments until observing a response
- Gaming: Attempts until winning
- Network reliability: Transmissions until successful delivery
Frequently Asked Questions
- What's the difference between geometric and binomial distribution?
- Binomial distribution counts the number of successes in a fixed number of trials, while geometric distribution counts the number of trials needed to get the first success. Geometric has a variable number of trials, binomial has a fixed number.
- What does the memoryless property mean?
- The memoryless property means that if you've had several failures, the probability of success on the next trial remains the same. Past failures don't change future probabilities in a geometric distribution.
- Can the geometric distribution model multiple successes?
- No, the standard geometric distribution only models the first success. For multiple successes, you would use the negative binomial distribution instead.
- Why is the minimum value k = 1?
- The geometric distribution starts at k = 1 because it represents the trial number where the first success occurs. The earliest possible first success is on the first trial, so k cannot be less than 1.
Related Calculators
statistics
Binomial Distribution Calculator