🎲 Geometric Distribution Calculator
Calculate the probability that the first success occurs on a specific trial or within a number of trials.
📊 Statistics📐 Formula included🧠 Practice questions✓ Free
Please enter valid values.
Formula and Explanation
Calculate the probability that the first success occurs on a specific trial or within a number of trials.
| Concept | Formula or rule |
|---|---|
| Geometric Distribution Calculator | P(X=k) = (1−p)ᵏ⁻¹p |
How to Use This Calculator
- Enter the sample, counts, probabilities, or summary statistics requested above.
- Keep paired lists in the same order and separate values with commas.
- Select the correct test direction or sample/population option when available.
- Press Calculate and interpret the result together with the supporting statistics.
Worked Example
Example: With p=0.25, the chance the first success occurs on trial 4 is 0.75³×0.25.
✓ P(X=4) ≈ 0.1055
Where This Is Used
School & coursework
Check homework, understand formulas, and verify step-by-step statistics exercises.
Research
Explore estimates, uncertainty, distributions, and relationships before using full statistical software.
Business & data analysis
Summarize variation, compare groups, and communicate data-driven findings clearly.
Common Mistakes to Avoid
⚠️
Starting k at zero
This page defines X as the trial number, so k starts at 1.
This page defines X as the trial number, so k starts at 1.
⚠️
Changing p between trials
The geometric model requires constant p.
The geometric model requires constant p.
⚠️
Confusing first success with number of successes
It models waiting time, not a count in fixed trials.
It models waiting time, not a count in fixed trials.
Practice Questions
1. If p=0.5, what is P(first success on trial 1)?
Answer: 0.5
2. If p=0.2, what is the expected trial number?
Answer: 5
3. What is the minimum trial number?
Answer: 1
Frequently Asked Questions
What is the expected trial number?▾
1/p.
Does the geometric distribution have memory?▾
It is memoryless under independent constant-probability trials.
What is P(X>k)?▾
(1−p)^k.