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Negative Binomial Distribution — result sheet
Calculate the probability of observing a selected number of failures before a specified number of successes.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: P(X=k) = C(k+r-1,k) p^r (1-p)^k, where X counts failures before the rth success.
The negative binomial model counts failures before a selected number of successes in independent identical Bernoulli trials. The ordering convention matters, so this page explicitly uses failures before the rth success.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Required successes r — minimum 1; maximum 10000
- Failures before r successes k — minimum 0; maximum 10000
- Success probability p — minimum 1.0E-9; maximum 1
Worked example
| Input | Value |
|---|---|
| Required successes r | 3 |
| Failures before r successes k | 2 |
| Success probability p | 0.5 |
Probability of 2 failures before the 3rd success is 0.1875.
Assumptions and limits
- Trials are independent with a constant success probability p.
- r is a positive integer and k is a nonnegative integer.
- The probability is calculated for one failure count and is not a fitted model or prediction of an external process.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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