Negative Binomial Probability Calculator
What Is the Negative Binomial Distribution?
The negative binomial distribution is a discrete probability distribution that counts the number of failures occurring before the r-th success in a sequence of independent Bernoulli trials. Each trial ends in success with probability p or failure with probability q = 1 − p, and the outcome of each trial does not affect any other.
It is used whenever the number of required successes is known in advance but the number of failures to reach that goal is uncertain. A manufacturing inspector who keeps testing items until finding 5 non-defective ones, a salesperson making calls until 3 conversions occur, a clinical trial enrolling patients until 10 eligible participants are found — all follow the negative binomial distribution.
Parameterization warning: Different textbooks and software packages define X differently. On this page, X counts failures before the r-th success (X = 0, 1, 2, …). Some sources define X as the trial number on which the r-th success occurs (X = r, r+1, r+2, …), so Xhere = Xother − r. Always check your source before using a table or calculator.
Negative Binomial Distribution Table
Select the PMF tab for exact probabilities P(X = x) or the CDF tab for cumulative probabilities P(X ≤ x). X is the number of failures before the r-th success. Each subtable below covers a fixed r value with success probabilities p = 0.10, 0.20, 0.25, 0.30, 0.40, 0.50, 0.60, 0.70, 0.75, 0.80, 0.90. Click any cell to load it into the calculator.
All values computed from P(X = x) = C(x + r − 1, x) × pr × (1 − p)x. X = number of failures before the r-th success. Values below 0.00005 shown as 0.0000. CDF values computed by summing PMF values from x = 0.
Negative Binomial Distribution Formula
The PMF of the negative binomial distribution — under the convention that X counts failures before the r-th success — is:
where C(x + r − 1, x) = (x + r − 1)! / (x! × (r − 1)!) is the binomial coefficient counting the number of ways to arrange x failures among the first x + r − 1 trials (the final trial is always the r-th success).
| Symbol | Meaning | Range | Example |
|---|---|---|---|
| X | Failures before r-th success | 0, 1, 2, … | X = 4 failures before 3rd sale |
| r | Number of successes required | 1, 2, 3, … | r = 3 sales to reach quota |
| p | Probability of success per trial | 0 < p < 1 | p = 0.30 conversion rate |
| q | Probability of failure per trial | q = 1 − p | q = 0.70 |
| C(n,k) | Binomial coefficient ("n choose k") | n! / (k!(n−k)!) | C(6,4) = 15 |
Expected Value and Variance
With r = 3 and p = 0.50: E(X) = 3(0.50)/0.50 = 3 expected failures; Var(X) = 3(0.50)/0.25 = 6.
How to Use the Negative Binomial Table
Reading the table is a five-step process. The examples that follow use real research contexts so the steps are concrete rather than abstract.
Step 1 — Confirm the Distribution Applies
Three conditions must hold: (1) each trial results in success or failure, (2) the success probability p stays constant across trials, and (3) trials are independent. If any condition fails, the negative binomial model does not apply. For situations where p changes over time, a different approach is needed.
Step 2 — Identify r, p, and x
r is the number of successes that must be achieved. p is the per-trial success probability. x is the quantity of interest — the number of failures before the r-th success occurs. Write these down before touching the table.
Step 3 — Choose PMF or CDF
Use the PMF table when the question asks for exactly x failures: "What is the probability of exactly 4 failures?" Use the CDF table when the question asks for at most x failures: "What is the probability of 4 or fewer failures?" For at least x failures, compute P(X ≥ x) = 1 − P(X ≤ x − 1) using the CDF.
Step 4 — Locate the Subtable and Cell
Navigate to the subtable for your r. Identify the column for your p. Find the row for x. The cell at that intersection gives P(X = x) on the PMF tab or P(X ≤ x) on the CDF tab.
Step 5 — Interpret the Result
A PMF value is a probability between 0 and 1. A CDF value is also a probability between 0 and 1, always at least as large as the corresponding PMF value. State the result in the language of the original problem. If the table does not contain your exact p value, use the calculator above for any p between 0 and 1.
Worked Examples Using the Table
Three complete examples below cover different probability types, different r values, and different applied contexts. Each shows every step from problem setup through table lookup to final interpretation.
Example 1 — Quality Control (Exact Probability)
Scenario: A factory tests electronic components one at a time. Each component is defective with probability 0.20 (so each is non-defective with p = 0.80). A quality engineer keeps testing until finding 3 non-defective components (r = 3). What is the probability that exactly 2 defective components appear before the 3rd good one (x = 2)?
| Step | Action | Result |
|---|---|---|
| 1 | Identify parameters | r = 3, p = 0.80, x = 2 |
| 2 | Choose table type | PMF — exact probability P(X = 2) |
| 3 | Locate subtable and column | r = 3 subtable, p = 0.80 column |
| 4 | Read table cell x = 2 | P(X = 2) = 0.1229 |
| 5 | Verify by formula | C(4,2) × 0.80³ × 0.20² = 6 × 0.512 × 0.04 = 0.1229 ✓ |
Interpretation: There is a 12.29% chance that exactly 2 defective components are found before the 3rd good one. In practice, the engineer can also check E(X) = 3(0.20)/0.80 = 0.75, confirming that fewer than 1 defective item is expected on average before the 3rd good component is found.
Example 2 — Sales Calls (Cumulative Probability)
Scenario: A salesperson has a 30% success rate on each cold call (p = 0.30). She needs to close 2 deals in a day (r = 2). Her manager wants to know: what is the probability she makes at most 4 unsuccessful calls before her 2nd sale — that is, P(X ≤ 4)?
| Step | Action | Result |
|---|---|---|
| 1 | Identify parameters | r = 2, p = 0.30, x ≤ 4 |
| 2 | Choose table type | CDF — cumulative P(X ≤ 4) |
| 3 | Locate subtable and column | r = 2 subtable, p = 0.30 column |
| 4 | Read CDF cell x = 4 | P(X ≤ 4) = 0.4718 |
Interpretation: The salesperson has about a 47% chance of closing her 2nd deal within 6 total calls (2 successes + at most 4 failures). The expected number of failed calls before her 2nd sale is E(X) = 2(0.70)/0.30 ≈ 4.67, so 4 failed calls is close to the expected value.
Example 3 — Clinical Screening (At Least Probability)
Scenario: A clinical trial requires 5 eligible patients (r = 5). Screening shows 40% of candidates meet eligibility criteria (p = 0.40). The research coordinator needs to know: what is the probability that at least 5 ineligible patients are screened before the 5th eligible one is found — P(X ≥ 5)?
| Step | Action | Result |
|---|---|---|
| 1 | Identify parameters | r = 5, p = 0.40, want P(X ≥ 5) |
| 2 | Convert to complement | P(X ≥ 5) = 1 − P(X ≤ 4) |
| 3 | Locate CDF cell x = 4, r = 5, p = 0.40 | P(X ≤ 4) = 0.2465 |
| 4 | Compute complement | P(X ≥ 5) = 1 − 0.2465 = 0.7535 |
Interpretation: There is a 75% chance the coordinator will screen at least 5 ineligible patients before enrolling the 5th eligible one. This informs planning: the expected number of ineligible screens is E(X) = 5(0.60)/0.40 = 7.5, and the standard deviation is √(Var(X)) = √(5 × 0.60 / 0.16) ≈ 4.33.
PMF vs CDF: Which Table Do You Need?
The two tabs of the table serve different question types. Choosing the wrong one is the most common student error.
P(X ≤ x) — CDF → "at most x failures" → use CDF tab
P(X < x) — → same as P(X ≤ x − 1) → use CDF tab with x − 1
P(X ≥ x) — → 1 − P(X ≤ x − 1) → use CDF tab with x − 1, then subtract from 1
P(X > x) — → 1 − P(X ≤ x) → use CDF tab with x, then subtract from 1
P(a ≤ X ≤ b) — → P(X ≤ b) − P(X ≤ a − 1) → two CDF lookups
Numerical check: r = 2, p = 0.50, x = 3
CDF: P(X ≤ 3) = 0.8125
P(X ≥ 3) = 1 − P(X ≤ 2) = 1 − 0.6875 = 0.3125
P(1 ≤ X ≤ 3) = P(X ≤ 3) − P(X ≤ 0) = 0.8125 − 0.2500 = 0.5625
Negative Binomial vs Binomial vs Geometric
These three distributions all model independent Bernoulli trials with fixed p, but they answer different questions. Choosing the right model is essential for a correct probability calculation.
| Feature | Binomial | Geometric | Negative Binomial |
|---|---|---|---|
| What is fixed? | Number of trials n | Number of successes r = 1 | Number of successes r |
| What is counted? | Successes in n trials | Failures before 1st success | Failures before r-th success |
| Parameters | n, p | p | r, p |
| Sample space | 0, 1, …, n | 0, 1, 2, … | 0, 1, 2, … |
| E(X) | np | (1 − p) / p | r(1 − p) / p |
| Special case | — | Neg. Binom. with r = 1 | Geometric when r = 1 |
| Typical question | "How many heads in 10 flips?" | "How many tails before first head?" | "How many tails before 3rd head?" |
| PMF formula | C(n,x) p^x q^(n−x) | p × q^x | C(x+r−1,x) p^r q^x |
Geometric as a special case
Setting r = 1 in the negative binomial PMF gives C(x,x) × p × q^x = p × q^x, which is the geometric PMF. This is why the r = 1 subtable in the negative binomial table above is identical to a geometric distribution table. See basic probability for the foundation that connects these distributions, and visit Statistics Fundamentals for the full discrete distribution collection.
Negative Binomial vs Poisson Distribution
Both distributions model counts of rare events, but they differ in structure and assumptions. The key distinction: the Poisson assumes a fixed rate λ with variance equal to its mean. The negative binomial has no such constraint — its variance always exceeds its mean, which makes it suited to overdispersed count data where the Poisson model fits poorly.
When to Use Negative Binomial
Trials are clearly sequential and you count failures before a fixed number of successes. Each trial is independent with the same p. The number of required successes r is known before the process begins.
When to Use Poisson
Events occur continuously in time or space at a constant average rate λ. Individual trials are not distinguishable. Examples include customer arrivals per hour, defects per meter of cable, or accidents per month.
Common Mistakes to Avoid
These errors appear repeatedly in student work and exam answers. Each one produces a wrong numerical answer, so checking for them explicitly is worthwhile.
Mistake 1 — Confusing X with the total number of trials
X on this page counts failures only. Total trials = X + r. If the problem says "the 3rd success occurs on the 8th trial," then x = 8 − 3 = 5 failures. Do not look up x = 8 in the table; look up x = 5.
Mistake 2 — Using PMF when CDF is needed (or vice versa)
"At most 4 failures" requires P(X ≤ 4) from the CDF, not P(X = 4) from the PMF. "Exactly 4 failures" requires the PMF. Read the question phrase carefully before selecting a tab.
Mistake 3 — Applying the model to dependent trials
If drawing without replacement from a finite population, p changes on each draw. In this case the hypergeometric distribution, not the negative binomial, is the correct model. The chi-square test and related procedures handle count data with dependence.
Mistake 4 — Ignoring software parameterization differences
R's dnbinom(x, size=r, prob=p) uses the failures-before-r-th-success convention — matching this table. SciPy's scipy.stats.nbinom.pmf(x, n=r, p=p) also uses this convention. Excel's NEGBINOM.DIST(x, r, p, cumulative) uses the same convention. However, some older references define x as total trials rather than failures — always verify before importing results.
Mistake 5 — Confusing geometric and negative binomial with r > 1
The geometric distribution applies only when waiting for the first success (r = 1). When r > 1, the geometric PMF gives the wrong answer. Always verify r before selecting a table.
Software Implementation
All major statistical software packages support the negative binomial distribution. The examples below compute P(X = 4) and P(X ≤ 4) with r = 3 and p = 0.40.
R
pnbinom(4, size = 3, prob = 0.40) # CDF: P(X ≤ 4) = 0.2465
size = r (number of successes); prob = p; x = number of failures.
Python (SciPy)
nbinom.pmf(4, n=3, p=0.40) # P(X = 4) = 0.0823
nbinom.cdf(4, n=3, p=0.40) # P(X ≤ 4) = 0.2465
n = r (successes); p = success probability; x = failures before r-th success.
Excel / Google Sheets
=NEGBINOM.DIST(4, 3, 0.40, TRUE) # CDF: 0.2465
Arguments: (x failures, r successes, p, cumulative TRUE/FALSE). Google Sheets uses the same syntax.
Practice Problems
These five problems cover the full range of probability types. Use the table above to look up probabilities, then verify with the calculator.
Problem 1 — Exact probability
A basketball player makes 60% of free throws (p = 0.60). What is the probability that she misses exactly 2 shots before making her 3rd free throw? [Answer: P(X = 2) for r = 3, p = 0.60]
Problem 2 — Cumulative probability
A production line passes 70% of items (p = 0.70). Inspection continues until 4 passing items are found (r = 4). What is the probability that at most 3 failing items are inspected before the 4th pass? [CDF tab, r = 4, p = 0.70, x = 3]
Problem 3 — At least probability
A hiring manager accepts 25% of applicants (p = 0.25). She needs 2 hires (r = 2). What is the probability of interviewing at least 6 applicants who are rejected before the 2nd hire? [P(X ≥ 6) = 1 − CDF(5), r = 2, p = 0.25]
Problem 4 — Range probability
A network engineer finds 50% of servers have a configuration error (so p = 0.50 of being error-free). She audits until finding 3 clean servers (r = 3). Find P(2 ≤ X ≤ 5). [CDF(5) − CDF(1), r = 3, p = 0.50]
Problem 5 — Expected value and comparison
A researcher tests drug candidates with a 20% success rate (p = 0.20). Five successes (r = 5) are needed before a trial can proceed. Calculate E(X) and Var(X). Then find P(X ≤ E(X)). What does this tell you about the shape of the distribution?
Symbol and Concept Glossary
| Term | Definition | Formula / Notation | Related Concept |
|---|---|---|---|
| Negative Binomial Distribution | Distribution of failures before r-th success | NB(r, p) | Binomial, Geometric |
| PMF | Probability Mass Function — exact probability P(X = x) | C(x+r−1,x)p^r q^x | CDF |
| CDF | Cumulative Distribution Function — P(X ≤ x) | Σ PMF(k), k=0 to x | PMF |
| Bernoulli Trial | Single independent trial with success probability p | P(success) = p | Binomial, Geometric |
| E(X) | Expected number of failures before r-th success | r(1−p)/p | Mean, Variance |
| Var(X) | Variance of failures before r-th success | r(1−p)/p² | E(X), Standard Deviation |
| Geometric Distribution | Special case NB(1, p) — failures before first success | p × (1−p)^x | Negative Binomial |
| Pascal Distribution | Alternative name for the negative binomial distribution | Same as NB(r, p) | Blaise Pascal |
References and Further Reading
DeGroot, M. H. & Schervish, M. J. (2012). Probability and Statistics (4th ed.). Addison-Wesley. Section 5.5 — Negative Binomial distribution, including derivation of the PMF, expected value, and variance. Standard graduate-level probability reference.
NIST/SEMATECH e-Handbook of Statistical Methods. Section 1.3.6.17: Negative Binomial Distribution. National Institute of Standards and Technology. itl.nist.gov — U.S. government statistical reference with the PMF formula, parameter definitions, and worked example.
Penn State STAT 414: Probability Theory. Lesson 11.3 — The Negative Binomial Distribution. Pennsylvania State University. online.stat.psu.edu — Open-access course notes covering the derivation, applications, and relationship to the geometric distribution.
Wackerly, D., Mendenhall, W. & Scheaffer, R. (2008). Mathematical Statistics with Applications (7th ed.). Cengage. Chapter 3 — Discrete random variables and their distributions, including the negative binomial as an extension of the geometric. Commonly used in upper-level statistics courses.
SciPy Documentation — scipy.stats.nbinom. docs.scipy.org — Official documentation for SciPy's negative binomial implementation, including parameter conventions and worked examples in Python.
Related Statistical Tables & Resources
Understanding the Negative Binomial Distribution
Why the Name "Negative Binomial"?
The name comes from its PMF formula. The binomial coefficient C(x + r − 1, x) can be written using the generalized binomial expansion with a negative exponent — hence "negative binomial." Blaise Pascal and Jacob Bernoulli both studied waiting-time distributions in the 17th century, which is why the distribution is also called the Pascal distribution in some references.
Shape of the Distribution
The negative binomial distribution is right-skewed when p is small (many failures expected before each success) and becomes more symmetric as p approaches 1. For large r, the Central Limit Theorem causes the distribution to approach normality. The variance always exceeds the mean — this overdispersion distinguishes it from the Poisson model, where variance equals the mean.
Connection to the Binomial Table
The negative binomial CDF is related to the binomial CDF: P(X ≤ x) in the negative binomial equals P(Y ≥ r) in the binomial with parameters n = x + r and p, where Y counts successes. This relationship lets you cross-check negative binomial probabilities using a binomial table when needed. The full derivation is covered in probability theory courses using the beta-binomial relationship.