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Negative Binomial Calculator

Compute the probability of the r-th success on trial k.

Introduction

Negative Binomial Calculator — Compute the probability of the r-th success on trial k. Enter Target successes (r), Success probability (p), Trial of r-th success (k) to get an instant, accurate result.

Formula

PMF: P(X=k) = C(k−1, r−1) × pʳ × (1−p)^(k−r), the probability that the r-th success occurs on the k-th trial.

Step-by-step

  1. Enter the Target successes (r).
  2. Enter the Success probability (p).
  3. Enter the Trial of r-th success (k).
  4. Click Calculate to see your result instantly.

Real-world example

Example: With Target successes (r) = 3, Success probability (p) = 0.5, Trial of r-th success (k) = 5, the Negative Binomial Calculator gives Probability: 0.1875.

Frequently Asked Questions

How does this relate to the geometric distribution?
The geometric distribution is a special case of the negative binomial where r = 1 (waiting for just the first success).
What does k represent here?
k is the trial number on which the r-th success occurs, so k must be greater than or equal to r.
What inputs does the Negative Binomial Calculator need?
You'll need to provide: Target successes (r), Success probability (p), Trial of r-th success (k). All fields use sensible defaults, so you can see a working example immediately and then adjust the values to match your own numbers.
How accurate is the Negative Binomial Calculator?
The Negative Binomial Calculator applies the formula exactly as calculated — PMF: P(X=k) = C(k−1, r−1) × pʳ × (1−p)^(k−r), the probability that the r-th success occurs on the k-th trial. — so results are precise for the inputs you provide. Accuracy depends on entering correct, realistic input values.
Is the Negative Binomial Calculator free to use?
Yes, the Negative Binomial Calculator is completely free, requires no signup, and runs instantly in your browser.

About the Negative Binomial Calculator

The Negative Binomial Calculator uses a real, verifiable formula — PMF: P(X=k) = C(k−1, r−1) × pʳ × (1−p)^(k−r), the probability that the r-th success occurs on the k-th trial. — so results are accurate every time, not an approximation.