Geometric Distribution Calculator
Calculate probabilities for geometric distribution - the number of failures before the first success
Calculate Geometric Distribution
Number of failures before the first success (non-negative integer)
Probability of success in a single trial (between 0 and 1)
Distribution Results
Formula used: P(X = x) = (1-p)^x × p
Where: x = 1 failures, p = 0.5 success probability
Interpretation: The probability of getting exactly 1 failures before the first success is 25.0000%
Probability Distribution Table
| Failures (x) | P(X = x) | Percentage | P(X ≤ x) |
|---|---|---|---|
| 0 | 0.500000 | 50.0000% | 0.500000 |
| 1 | 0.250000 | 25.0000% | 0.750000 |
| 2 | 0.125000 | 12.5000% | 0.875000 |
| 3 | 0.062500 | 6.2500% | 0.937500 |
| 4 | 0.031250 | 3.1250% | 0.968750 |
| 5 | 0.015625 | 1.5625% | 0.984375 |
| 6 | 0.007813 | 0.7813% | 0.992188 |
Example: Dice Rolling
Problem
Scenario: Rolling a die until getting a 6
Success probability (p): 1/6 ≈ 0.1667
Question: What's the probability of getting exactly 1 failure (roll on the 2nd attempt)?
Solution
P(X = 1) = (1-p)¹ × p
P(X = 1) = (1-1/6)¹ × 1/6
P(X = 1) = (5/6) × (1/6)
P(X = 1) = 5/36 ≈ 0.1389 = 13.89%
Distribution Properties
PMF
P(X=x) = (1-p)^x × p
Probability mass function
Mean
μ = (1-p)/p
Expected number of failures
Variance
σ² = (1-p)/p²
Measure of variability
Memoryless
P(X=n+k|X≥n) = P(X=k)
Key property of geometric distribution
Common Examples
Rolling dice until getting a specific number
Flipping coins until getting heads
Shooting arrows until hitting the target
Quality control testing until finding a defect
Medical tests until positive result
Network packet transmission until success
Understanding Geometric Distribution
What is Geometric Distribution?
The geometric distribution models the number of failures that occur before the first success in a sequence of independent Bernoulli trials. Each trial has two possible outcomes (success or failure) with the same probability of success p.
Key Characteristics
- •Discrete probability distribution
- •Memoryless property
- •Support: x = 0, 1, 2, 3, ...
- •Right-skewed distribution
Mathematical Formulas
Probability Mass Function
P(X = x) = (1-p)^x × p
where x = 0, 1, 2, ... and 0 < p ≤ 1
Cumulative Distribution Function
P(X ≤ x) = 1 - (1-p)^(x+1)
Distribution Parameters
- Mean: μ = (1-p)/p
- Variance: σ² = (1-p)/p²
- Standard Deviation: σ = √[(1-p)/p²]
Memoryless Property
One of the most important characteristics of the geometric distribution is its memoryless property. This means that the probability of achieving success in future trials is independent of the number of failures already observed. Mathematically:
P(X = n + k | X ≥ n) = P(X = k)
The probability of waiting k more trials given you've already waited n trials equals the probability of waiting k trials from the start.