Negative Binomial Calculator Formula

Understand the math behind the negative binomial calculator. Each variable explained with a worked example.

Use the Negative Binomial Calculator

Formulas Used

P(X = k)

probability = coeff * pow(p, r) * pow(1 - p, k - r)

Expected Trials

expected = r / p

Variance

variance_val = r * (1 - p) / pow(p, 2)

Variables

VariableDescriptionDefault
rTarget Successes (r)3
kTotal Trials (k)8
pSuccess Probability (p)0.4
coeffDerived value= factorial(k - 1) / (factorial(r - 1) * factorial(k - r))calculated

How It Works

How to Calculate Negative Binomial Probability

Formula

P(X = k) = C(k-1, r-1) * p^r * (1-p)^(k-r)

The negative binomial distribution models the number of trials needed to achieve exactly r successes. The last trial must be a success (the r-th), and the preceding k-1 trials must contain exactly r-1 successes. The expected number of trials is r/p.

Worked Example

A salesperson closes 40% of pitches. What is the probability that the 3rd sale happens on the 8th pitch?

r = 3k = 8p = 0.4
  1. 01Need r=3 successes on trial k=8
  2. 02C(7,2) = 21
  3. 03p^r = 0.4^3 = 0.064
  4. 04(1-p)^(k-r) = 0.6^5 = 0.07776
  5. 05P(X=8) = 21 * 0.064 * 0.07776 ≈ 0.10450
  6. 06Expected trials = 3 / 0.4 = 7.5

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