Calculator
Example Data Table
| Required successes (r) | Success probability (p) | Failures (k) | P(X = k) | P(X ≤ k) |
|---|---|---|---|---|
| 3 | 0.40 | 0 | 0.06400000 | 0.06400000 |
| 3 | 0.40 | 1 | 0.11520000 | 0.17920000 |
| 3 | 0.40 | 2 | 0.13824000 | 0.31744000 |
Formula Used
Definition: X counts failures before the r-th success when each trial has the same success probability p.
Exact probability: P(X = k) = C(k + r - 1, k) × pr × (1 - p)k
Cumulative probability: P(X ≤ k) = Σ P(X = i), for i from 0 to k
Range probability: P(a ≤ X ≤ b) = Σ P(X = i), for i from a to b
Mean: r(1 - p) / p
Variance: r(1 - p) / p2
Standard deviation: √Variance
Mode: floor(((r - 1)(1 - p)) / p), when r > 1
Skewness: (2 - p) / √(r(1 - p))
How to Use This Calculator
- Enter the number of required successes.
- Enter the success probability for one trial.
- Enter the observed number of failures you want to test.
- Set a lower and upper bound for an inclusive probability range.
- Choose the maximum x value for the distribution table.
- Set decimal places for the output.
- Press the calculate button to view results above the form.
- Use the CSV or PDF buttons to export the summary and table.
About the Negative Binomial Distribution
What This Calculator Solves
The negative binomial distribution models repeated independent trials. It tracks how many failures happen before a fixed number of successes appears. This makes it useful when one success is not enough. Many real processes need several successful outcomes before a task is complete.
Why Analysts Use It
This calculator helps measure exact probability, cumulative probability, and interval probability. It also reports the mean, variance, standard deviation, mode, and skewness. These values describe the center, spread, and shape of the distribution. They help analysts understand uncertainty before making decisions.
Common Real Applications
Quality teams use this model to study defects before a target number of good units is produced. Support teams use it to estimate failed contacts before enough customer responses arrive. Reliability analysts use it for repeated testing. Marketing teams can apply it to campaign attempts before reaching a goal number of conversions.
How the Parameters Work
The parameter r is the number of required successes. The parameter p is the probability of success on each trial. The variable X counts failures before the r-th success. When p gets smaller, the expected number of failures usually rises. When r gets larger, the distribution shifts right because more successful outcomes are needed.
Why Exporting Matters
Export options make this tool practical for reporting and review. You can save summary metrics and the full probability table as CSV or PDF. That helps with audits, classroom work, presentations, and statistical documentation. Clean exported tables also support team collaboration and model checking.
Best Practice for Interpretation
Always confirm that trials are independent and that the success probability stays constant. Those assumptions matter. If the process changes over time, the negative binomial model may no longer fit well. In that case, estimates can drift and decisions can become weaker.
FAQs
1. What does this calculator measure?
It measures probabilities for the number of failures that occur before reaching a fixed number of successes in repeated independent trials.
2. What does r represent?
r is the target number of successes. The calculator keeps counting trials until that success target is reached.
3. What does p represent?
p is the probability of success on one trial. It must stay constant across all trials for the model to fit correctly.
4. What is k in the results?
k is the observed number of failures before the required successes occur. The exact probability is calculated at that value.
5. When should I use a negative binomial model?
Use it when trials are independent, success probability stays unchanged, and you care about failures before a fixed success count.
6. What is the difference between exact and cumulative probability?
Exact probability gives the chance of one specific failure count. Cumulative probability adds all probabilities from zero up to that count.
7. Why does the table stop at a selected upper x?
The upper x value limits the output table size. It keeps the page readable while still showing a useful probability range.
8. Can I use the exports in reports?
Yes. The CSV and PDF exports are useful for assignments, audit files, dashboards, and management summaries.