Geometric Distribution Calculator
Free geometric distribution calculator for probability calculations. Calculate exact, cumulative, and at-least probabilities with step-by-step solutionsfor statistics and probability theory. Perfect for students learning discrete probability distributions.
Last updated: December 15, 2024
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Geometric Distribution Results
Probability:
0.1029
Exact Probability
Formula: P(X = k) = (1-p)^(k-1) × p
Cumulative Probability:
0.7599
Mean (Expected Value):
3.333
Variance:
7.778
Step-by-Step Solution:
Geometric Distribution Tips:
- • P(X = k) = (1-p)^(k-1) × p for exact probability
- • P(X ≤ k) = 1 - (1-p)^k for cumulative probability
- • P(X ≥ k) = (1-p)^(k-1) for at least probability
- • Mean = 1/p, Variance = (1-p)/p²
- • Models number of trials until first success
Geometric Distribution Types
Formula
P(X = k) = (1-p)^(k-1) × p
Probability of exactly k trials until first success
Formula
P(X ≤ k) = 1 - (1-p)^k
Probability of success within k trials or fewer
Formula
P(X ≥ k) = (1-p)^(k-1)
Probability of needing at least k trials
Example
Items inspected until first defect
p = defect rate, k = inspection number
Example
Attempts until first win
p = win probability, k = attempt number
Example
Time until first system failure
p = failure rate, k = time period
Quick Example Result
Geometric distribution with p = 0.3, k = 4:
Exact Probability
0.1029
Cumulative
0.7599
Mean
3.333
How to Calculate Geometric Distribution
Geometric distribution is a fundamental discrete probability distribution that models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. Understanding this distribution is crucial for statistics, quality control, and decision-makingin various fields where waiting time until success is important.
The Geometric Distribution Process
This systematic approach ensures accurate geometric distribution calculations for any scenario.
Geometric Distribution Formulas
The geometric distribution has three main formulas: P(X = k) = (1-p)^(k-1) × p for exact probability, P(X ≤ k) = 1 - (1-p)^k for cumulative probability, and P(X ≥ k) = (1-p)^(k-1) for at-least probability. The distribution is characterized by its mean μ = 1/p and variance σ² = (1-p)/p². The memoryless property means that the probability of success in future trials doesn't depend on past failures.
- Exact Probability: P(X = k) = (1-p)^(k-1) × p
- Cumulative Probability: P(X ≤ k) = 1 - (1-p)^k
- At Least Probability: P(X ≥ k) = (1-p)^(k-1)
- Mean: μ = 1/p
- Variance: σ² = (1-p)/p²
Sources & References
- Introduction to Probability - Joseph K. Blitzstein, Jessica HwangComprehensive coverage of discrete probability distributions including geometric
- Statistical Inference - George Casella, Roger L. BergerAdvanced statistical methods and distribution theory
- Khan Academy - Geometric DistributionVideo tutorials and practice problems on discrete probability distributions
Need help with other probability topics? Check out our percentage calculator and mixed number calculator.
Get Custom Calculator for Your PlatformGeometric Distribution Example
Given Parameters:
Success probability (p) = 0.3
Number of trials (k) = 4
Calculation type = Exact probability
Solution Steps:
- Step 1: Given parameters
- Success probability (p) = 0.3
- Number of trials (k) = 4
- Step 2: Apply geometric distribution formula
- P(X = k) = (1-p)^(k-1) × p
- P(X = 4) = (1-0.3)^(4-1) × 0.3
- P(X = 4) = 0.700^3 × 0.3
- P(X = 4) = 0.343 × 0.3 = 0.1029
- Step 3: Calculate cumulative probability
- P(X ≤ k) = 1 - (1-p)^k
- P(X ≤ 4) = 1 - (1-0.3)^4
- P(X ≤ 4) = 1 - 0.700^4 = 1 - 0.2401 = 0.7599
Final Results:
Exact Probability
0.1029
Cumulative Probability
0.7599
Mean
3.333
Variance
7.778
Quality Control
p = 0.05 defect rate, k = 10 inspections
P(X = 10) = 0.0315
Game Theory
p = 0.2 win probability, k = 5 attempts
P(X = 5) = 0.0819
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