NEGBINOMDIST in Apple Numbers: Failures Before Successes

Use NEGBINOMDIST when the experiment continues until a fixed number of successes. The number of failures is random. Examples include retries until a target number of independently completed jobs or calls until a fixed number of independently successful outcomes; the same per-attempt probability must remain plausible. This differs from BINOMDIST, which fixes the total number of attempts and counts successes. For successes r≥1, failures f≥0, and constant success chance 0<p<1, the standard count probability is:

P(F=f before r successes) = COMBIN(f+r-1, f) × p^r × (1-p)^f

The combinatorial term counts the ways the first f+r−1 attempts can contain r−1 successes and f failures; the final attempt is the r-th success. That reasoning is an independent mathematical derivation, not a claim that Numbers evaluates COMBIN identically for every input. [A1, C1]

=NEGBINOMDIST(f-num, s-num, prob-success)

For a meaningful count distribution enter whole nonnegative failure counts and whole positive success targets. Apple’s article gives numeric bounds but does not establish a rounding or truncation rule for fractional f-num or s-num; don’t infer one from another spreadsheet. The endpoints p=0 and p=1 are excluded by Apple’s published bounds, even where a mathematical limit might be definable. There is no cumulative TRUE/FALSE switch or optional argument in this function’s documented signature. [A1]

Create a Numbers table named TrialsToTarget. Add header labels at B1 Failures f, C1 Probability at f, E1 Success target r, and F1 Success chance p. Put 3 in E2 and 0.4 in F2; put the whole numbers 0,1,2,3,4,5,6,7 in B2:B9. In C2 enter the formula below, then fill C2 through C9:

=NEGBINOMDIST(B2,$E$2,$F$2)

The $ keeps the success target and probability anchored as the failure count changes. In another table, the equivalent parameter cells would need their proper qualified references, such as TrialsToTarget::E2. This is Numbers notation—not an Excel sheet reference. [A1, A3]

These are independent probabilities, not cumulative probabilities. Adding just f=0..7 cannot equal one because larger failure counts remain possible. Expected values came from both a combinatorial formula and a separate SciPy implementation, not from Apple Numbers execution. [C1]

1. Probability of exactly two failures before three successes

=NEGBINOMDIST(2,3,0.4)

Expected 0.13824 (COMBIN(4,2)×0.4³×0.6²). It is important that the last (fifth) attempt is the third success. BINOMDIST(3,5,0.4,FALSE) asks a different fixed-five-attempts question. [A1, A6, C1]

2. Probability of at most two failures

NEGBINOMDIST lacks a cumulative-mode argument. Sum the relevant point probabilities yourself:

=NEGBINOMDIST(0,3,0.4)+NEGBINOMDIST(1,3,0.4)+NEGBINOMDIST(2,3,0.4)

Expected 0.31744, the sum of 0.064 + 0.1152 + 0.13824. With the table, =SUM(C2:C4) represents the same sum. Do not replace the last argument by TRUE: the third argument is a probability, not a mode selector. [A1, A6, C1]

3. More than two failures before hitting the target

=1-(NEGBINOMDIST(0,3,0.4)+NEGBINOMDIST(1,3,0.4)+NEGBINOMDIST(2,3,0.4))

Expected 0.68256, the complement of the ≤2-failure event. This assumes the negative-binomial model is appropriate and completion eventually occurs with positive per-trial success chance. [C1]

4. Interpret a high-success-rate process correctly

Apple’s published example =NEGBINOMDIST(3,68,0.95) gives 0.20913174716192. The count 68 is the success target, not the total number of opportunities. To estimate a retry budget in an actual process, validate the independence and constant-p assumptions; correlated outcomes or changing customer mix can make a seemingly precise probability unhelpful. [A1]

Apple’s help does not enumerate a universal error token, overflow threshold, rounding behavior or numerical-precision limits for these cases. Label them invalid/unsupported assumptions rather than promising a specific Excel-style error. Tiny probabilities can display as rounded zero in a narrow number-format cell. [A1, C1]

Is this a cumulative negative-binomial function? No: Apple’s function returns the probability of the specified failure count, with no cumulative mode. Sum the required failure-count probabilities to make your own cumulative value.

What is counted as success? Whatever single event you define as success, provided the model uses a stable probability per independent attempt; s-num is the required number of successes, not the successes already seen.

Can I copy the syntax from an application that calls the function NEGBINOM.DIST? Do not assume the spelling or cumulative options match. The English Apple canonical name documented here is NEGBINOMDIST, with no punctuation and three arguments. [A1]

• BINOMDIST fixes the number of independent trials and counts successes; it is not the same stopping rule. [A6]
• CRITBINOM returns a binomial count quantile, not a negative-binomial mass. [A6]

Apple’s English Formulas and Functions Help, edition 15.4, documents the function and its parameter order. Apple’s current Mac Numbers version history describes release 15.4, but does not identify this particular function among the features newly introduced in that release. The 15.4 documentation therefore cannot establish the earliest version that supports it. Apple currently states that Numbers 15.4 for Mac requires macOS 26 or later and Numbers 15.4 for iPad requires iPadOS 26 or later; availability, edge-case evaluation and precision on an actual Mac, iPhone, iPad or iCloud session have not been tested here. Do not treat Excel, Google Sheets or Python behavior as proof of Numbers compatibility. [A1, A4, A5]

The formulas here use the English decimal point and comma-separated arguments. Apple states that in a setting using the decimal comma you must separate function arguments with semicolons: for example, =NEGBINOMDIST(2;3;0,4) rather than =NEGBINOMDIST(2,3,0.4). This is a separator rule; it does not verify localized translations of function names or every regional input format. Apple says function names are case-insensitive. [A2]

Numbers uses named tables within sheets, with table-qualified cell references. A formula can refer to B2 in its own table, or to an explicitly qualified cell such as Model::B2 in another table, according to Apple’s cell-reference guidance. $ anchors hold fixed cells when filling formulas. [A3]

• [A1] Apple, NEGBINOMDIST, Formulas and Functions Help 15.4, https://support.apple.com/guide/functions/negbinomdist-ffa5aa5269/15.4/web/1.0. Checked 2026-10-13 UTC. Establishes function name, argument order, documented types/modes/bounds and Apple’s example; it does not certify historical app versions, native test outcomes or undocumented coercion.
• [A2] Apple, Functions overview, documentation edition 15.4. Checked 2026-10-13 UTC; establishes case-insensitive names, parenthesized arguments, required-versus-optional tokens and decimal-comma semicolon separator rule.
• [A3] Apple, Refer to cells in formulas, documentation edition 15.4. Checked 2026-10-13 UTC; establishes cell/range notation, cross-table :: qualifiers and Numbers-specific table references.
• [A4] Apple, What’s new in Numbers on Mac, checked 2026-10-13 UTC; version history describes Numbers 15.4 and macOS requirements. It does not individually date the introduction of this function.
• [A5] Apple, What’s new in Numbers on iPad and Formulas and Functions Help, checked 2026-10-13 UTC; establishes the current broader documentation/version context without establishing an individual iPhone/iCloud regression result.
• [A6] Apple, related references listed below are directly linked to their corresponding official 15.4 function help and checked 2026-10-13 UTC. They explain comparisons, not proof of another application’s behavior.
• [C1] Independent mathematical verification, 2026-10-13 UTC: local Python standard-library math evaluated against separately implemented open-source SciPy distribution calculations. Expected results are not native Numbers outputs or screenshots.

Native-validation limitations: No Numbers execution or screenshots were performed. Historical device compatibility, localized behavior and undocumented coercion/precision require testing on the intended platform.