🎯 Negative Binomial Calculator
Calculate the probability of a specified number of failures before a target number of successes.
📊 Statistics📐 Formula included🧠 Practice questions✓ Free
Please enter valid values.
Formula and Explanation
Calculate the probability of a specified number of failures before a target number of successes.
| Concept | Formula or rule |
|---|---|
| Negative Binomial Calculator | P(F=k) = C(k+r−1,k)(1−p)ᵏpʳ |
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: Find the probability of 4 failures before the 3rd success when p=0.4.
✓ P(F=4) = C(6,4)×0.6⁴×0.4³
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
⚠️
Mixing parameterizations
Some textbooks count total trials instead of failures.
Some textbooks count total trials instead of failures.
⚠️
Using k as successes
Here k is failures before the rth success.
Here k is failures before the rth success.
⚠️
Changing p across trials
The model assumes constant p.
The model assumes constant p.
Practice Questions
1. What special case occurs when r=1?
Answer: Geometric distribution.
2. If p=1, how many failures are expected?
Answer: 0
3. Does this model use independent trials?
Answer: Yes.
Frequently Asked Questions
How is it related to the geometric distribution?▾
Geometric is the special case r=1.
What is the expected number of failures?▾
r(1−p)/p.
Can k be zero?▾
Yes, all required successes may occur immediately.