Negative Binomial Distribution
Print PageA Negative Binomial Distribution Calculator (also known as a Pascal Distribution Calculator, r-th Success Probability Utility, Discrete Cumulative CDF Analyzer, or Bernoulli Multiple Waiting Time Calculator) computes exact single-point probabilities (P(X = k) = &binom;{k - 1}{r - 1} · pr · (1 - p)k - r for total trials k, or P(Y = y) = &binom;{y + r - 1}{y} · pr · (1 - p)y for total failures y = k - r), lower cumulative probabilities (P(X ≤ k)), upper cumulative probabilities (P(X ≥ k)), expected mean trials (μk = r ÷ p), expected mean failures (μy = r · [1 - p] ÷ p), variance (σ2 = r · [1 - p] ÷ p2), and standard deviation for obtaining r successes in independent Bernoulli trials with constant success probability p.
In enterprise sales team quota planning, bug hunting in software engineering, and clinical medical trial recruiting, the negative binomial distribution models the total number of attempts or failures required to achieve a target number of r successes.
Our free online Negative Binomial Distribution Calculator provides instant calculations across all discrete waiting parameters:
- Probability Mass Function (PMF for Total Trials k):
P(X = k) = &binom;{k - 1}{r - 1} · pr · (1 - p)k - r(fork = r, r+1, r+2...). - Alternative PMF (For Number of Failures y = k – r):
P(Y = y) = &binom;{y + r - 1}{y} · pr · (1 - p)y(fory = 0, 1, 2...). - Expected Mean Total Trials (μk):
μk = r ÷ p. - Expected Mean Failures (μy):
μy = r · ( 1 - p ) ÷ p. - Variance (σ2):
σ2 = r · ( 1 - p ) ÷ p2. - Standard Deviation (σ):
σ = √[ r · ( 1 - p ) ] ÷ p. - Geometric Special Case (r = 1): When
r = 1, the Negative Binomial distribution reduces exactly to the Geometric DistributionP(X = k) = (1 - p)k-1 · p.
Master Negative Binomial Probability Reference Table (r = 3 Closed Deals, p = 0.20 Win Rate)
The table below displays single-point probabilities P(X = k), lower cumulative probabilities P(X ≤ k), upper cumulative probabilities P(X ≥ k), and expected values for closing r = 3 enterprise B2B sales contracts with p = 0.20 (20% meeting win rate):
| Total Trials (k Meetings Conducted) | PMF Combination Formula | Exact P(X = k) | Lower Cumulative P(X ≤ k) | Upper Cumulative P(X ≥ k) | Sales Campaign Milestone |
|---|---|---|---|---|---|
| k = 3 Meetings (Minimum) | &binom;{2}{2} · (0.20)3 · (0.80)0 | 0.8000% (0.0080) | 0.8000% | 100.0000% | Perfect 3-for-3 Sweep |
| k = 5 Meetings | &binom;{4}{2} · (0.20)3 · (0.80)2 | 3.0720% (0.0307) | 5.7920% | 97.2800% | Early Fast Quota Success |
| k = 10 Meetings (10-Meeting Benchmark) | &binom;{9}{2} · (0.20)3 · (0.80)7 | 6.0398% (0.0604) | 32.2200% (32.22%) | 73.8198% | 32.2% Probability Closed Within 10 |
| k = 15 Meetings (μk Expected Mean) | &binom;{14}{2} · (0.20)3 · (0.80)12 | 4.9818% (0.0498) | 57.4385% (57.44%) | 47.5433% | Expected Average Campaign Length |
| k = 20 Meetings | &binom;{19}{2} · (0.20)3 · (0.80)17 | 3.0848% (0.0308) | 77.1936% (77.19%) | 25.8912% | High Confidence 77.2% Threshold |
Step-by-Step B2B Sales Campaign Calculation (r = 3, p = 0.20)
To determine how many pitch meetings an enterprise sales team must conduct to achieve their quarterly quota of r = 3 closed contracts with a p = 0.20 win rate:
Step 1 (Calculate Expected Mean Meetings μ_k): μk = r ÷ p = 3 ÷ 0.20 = 15.0 meetings
Step 2 (Calculate Expected Lost Deals μ_y): μy = 3 · (1 - 0.20) ÷ 0.20 = 3 · 0.80 ÷ 0.20 = 2.40 ÷ 0.20 = 12.0 lost deals
Step 3 (Calculate Variance σ^2): σ2 = 3 · 0.80 ÷ (0.20)^2 = 2.40 ÷ 0.04 = 60.0 &implies; σ = √60.0 = 7.746 meetings
Step 4 (Calculate Exact 10-Meeting Success Probability P[X = 10]): P(X = 10) = &binom;{9}{2} · (0.20)^3 · (0.80)^7 = 36 · 0.008 · 0.209715 = 0.060398 ≈ 6.04%
Step 5 (Calculate Probability of Closing 3 Deals Within 10 Meetings P[X ≤ 10]): P(X ≤ 10) = ∑ P(X = 3..10) = 32.22%
Thus, the sales team expects to conduct 15 meetings to close 3 contracts, with a 32.22% probability of achieving their goal in 10 or fewer meetings.
Discrete Models Comparison: Negative Binomial vs. Geometric vs. Binomial vs. Poisson
Below is a comparative reference chart detailing when to use the Negative Binomial distribution versus related probability models:
| Probability Model | Target Variable Being Modeled | Fixed Input Condition | Primary Practical Application |
|---|---|---|---|
| Negative Binomial (Pascal) | Trial number k of the r-th success (r ≥ 1) | Fixed r successes | Closing r sales deals, finding r defective units. |
| Geometric Distribution | Trial number k of the VERY FIRST success (r = 1) | Fixed r = 1 success | First success waiting times (Memoryless). |
| Binomial Distribution | Number of successes k in a fixed series of n trials | Fixed n trials | Quality control defects per fixed batch. |
| Poisson Distribution | Discrete event counts in a fixed continuous interval | Fixed rate λ | Website hits per hour, call center arrivals. |
History & Mathematics: 1713 Montmort to 1714 Blaise Pascal
1713 Pierre Rémond de Montmort & Problem of Points
In 1713, French mathematician Pierre Rémond de Montmort published early solutions to waiting times for multiple successes in Essay d’analyse sur les jeux de hazard.
1714 Blaise Pascal & The Pascal Distribution
In 1714, French mathematician Blaise Pascal and Abraham de Moivre formalized the mathematical formula for obtaining r successes in independent trials, giving rise to the term Pascal distribution.
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Frequently Asked Questions (FAQ)
What is the Negative Binomial Distribution formula?
The single-point PMF formula for total trials k is P(X = k) = &binom;{k - 1}{r - 1} · pr · (1 - p)k - r.
What is the difference between Geometric and Negative Binomial distributions?
The Geometric distribution models the number of trials until the first success (r = 1), whereas the Negative Binomial distribution generalizes this for any number of r successes (r ≥ 1).
What is the expected mean number of trials for a Negative Binomial distribution?
The expected mean total trials required to achieve r successes is μk = r ÷ p.