How to Use Negative Binomial Distribution Calculator
The Negative Binomial Distribution Calculator is straightforward once you understand which definition applies to your problem.
- Select Mode — Choose "By trial number k" if you want to find the probability that the r-th success occurs on exactly the k-th trial. Choose "By failure count x" if you want to find the probability that exactly x failures occur before the r-th success.
- Number of Successes (r) — Enter the target count of successes. Must be a positive integer.
- Probability of Success (p) — Enter the probability of success on a single trial (between 0 and 1).
- Trial Number k or Failure Count x — Enter the value matching your selected mode.
- Read Results — The Negative Binomial Distribution Calculator outputs exact probability, cumulative probability P(X ≤ k), right-tail probability P(X ≥ k), expected value, and variance.
Formula & Theory — Negative Binomial Distribution Calculator
The Negative Binomial Distribution Calculator implements the classic PMF. In trial-count form, where X = k is the trial on which the r-th success occurs:
P(X = k) = C(k − 1, r − 1) · p^r · (1 − p)^(k − r) for k ≥ r
In failure-count form, where Y = x is the number of failures before the r-th success (Y = X − r):
P(Y = x) = C(x + r − 1, x) · p^r · (1 − p)^x for x ≥ 0
| Symbol | Meaning |
|---|---|
| r | Number of successes required |
| p | Probability of success per trial |
| k | Trial number on which r-th success occurs |
| x | Number of failures before r-th success |
| C(a, b) | Binomial coefficient |
The Negative Binomial Distribution Calculator also computes summary statistics:
Mean = r / p
Variance = r · (1 − p) / p²
Cumulative and Right-Tail Probabilities
Beyond the point probability, the Negative Binomial Distribution Calculator computes P(X ≤ k) by summing the PMF from r up to k, and P(X ≥ k) as 1 − P(X ≤ k − 1). These are essential for quality control thresholds and hypothesis testing.
Use Cases for Negative Binomial Distribution Calculator
The Negative Binomial Distribution Calculator is widely applicable across statistics and industry:
- Statistics & Probability Education — Students use the Negative Binomial Distribution Calculator to verify textbook exercises and build intuition about over-dispersed count data.
- Quality Control — A factory inspector can use the Negative Binomial Distribution Calculator to find the probability of inspecting exactly k items before finding r defects.
- Marketing & Conversion — Estimate how many ad impressions are needed before r conversions occur. The Negative Binomial Distribution Calculator quantifies campaign efficiency.
- Game Drop Rates (Gacha) — The Negative Binomial Distribution Calculator models the number of attempts needed to collect r copies of a rare item with probability p per pull.
- Clinical Trials — Researchers use the Negative Binomial Distribution Calculator to plan how many trial participants are needed before observing r adverse events.
- Epidemiology — Model disease transmission clusters where the Negative Binomial Distribution Calculator accounts for extra variance compared to a Poisson model.
- Sports Analytics — Calculate the probability that a team achieves its r-th win on exactly game k of a series using the Negative Binomial Distribution Calculator.