NEGBINOMDIST in Zoho Sheet: Failures Before a Fixed Number of Successes

NEGBINOMDIST returns the probability of observing exactly x failures before the r-th success in repeated independent trials where each trial has the same success probability p. Zoho’s documented syntax is NEGBINOMDIST(x; r; p). For example, if a fictional process succeeds independently 40% of the time and needs three successes, =NEGBINOMDIST(2;3;0.4) predicts approximately 0.13824, or 13.824%, that exactly two failures occur before the third success. This is an independently calculated negative-binomial probability, not a native Zoho execution. Z1

The outcome being counted here is failures, not total attempts. If three successes occur after two failures, then the total number of attempts is five. That conversion prevents a frequent off-by-r error. Another common mix-up is with BINOMDIST, which fixes the total number of trials and counts successes; NEGBINOMDIST fixes the target number of successes and counts preceding failures.

NEGBINOMDIST(x; r; p)

All three arguments are required and the Zoho reference documents no optional cumulative argument and no defaults. The vendor’s stated #NUM! conditions include p outside [0,1], x<0 and r<1. However, whether the function treats fractional x/r by rejection, truncation or another rule is not established in the Zoho article. Use whole numbers and keep in mind that boundary p=0 makes an eventual success impossible, even though the Help error line explicitly excludes only p outside the range. Avoid predicting native boundary behaviour without a test. Z1

For 0<p<1 and integers r≥1, x≥0, the mass function is P(X=x)=C(x+r−1,r−1) p^r (1−p)^x, where C is a combinations count. In this model the expected failures are r(1−p)/p and the variance is r(1−p)/p². These are independent mathematical identities, not undocumented Zoho-specific options.

Suppose you are studying a hypothetical quality test with success probability p=0.4 and need r=3 successful attempts. Put headers in A1:E1: Failures, Target successes, Success probability, Exact probability, Cumulative probability. Fill A2:A8 with 0–6, B2:B8 with 3, and C2:C8 with 0.4. Enter =NEGBINOMDIST(A2;B2;C2) in D2, copy through D8. Enter =SUM($D$2:D2) in E2 and copy to E8 to create a separate cumulative calculation.

The exact probability for x=2 is 0.13824. Column E adds probabilities from zero through x and therefore measures a different event. For example, the cumulative probability of at most two failures is ≈ 0.31744. The complement, probability of three or more failures, is =1-SUM(D2:D4) ≈ 0.68256. This complement is meaningful only because the cumulative sum includes every possible failure count below the threshold; a truncated table does not add to one.

Verify one row without trusting the function

For x=2, r=3, p=0.4, the combinatorial factor is C(4,2)=6, so 6×(0.4)^3×(0.6)^2 = 6×0.064×0.36 = 0.13824. A proposed separate spreadsheet check is =COMBIN(4;2)*0.4^3*0.6^2; the result is independently derived. The existence and exact syntax of COMBIN should be checked in the target Zoho sheet before labelling this a tested native cross-check. When parameter values are not integers, this simple combinations interpretation is invalid without a more advanced extension.

Total attempts: Add the target successes to failures. In the example an outcome x=5 means five failed and three successful trials, or eight total attempts until the third success. To model “finishes within five attempts,” allow at most x=5−3=2 failures; the probability is 0.31744. Simply evaluating NEGBINOMDIST(5;3;0.4) would be answering an entirely different question.

Expected workload: The mean failures to reach r=3 successes at p=0.4 is 3×0.6/0.4 = 4.5 failures. Expected total trials are r/p=7.5. Fractional expectations describe averages over many runs, not an impossible half of a single attempt. The standard deviation of the failure count is sqrt(3×0.6/0.4²) ≈ 3.354101966250. Actual outcomes can be highly dispersed; planning only from the expected value can understate capacity requirements.

At most 6 failures: Summing every exact probability from x=0 through x=6 gives approximately 0.768212992, leaving a 23.18% chance of more than six failures in the idealized model. A proposed spreadsheet check is =SUM(D2:D8); there is no cumulative toggle documented for Zoho’s NEGBINOMDIST. A different cutoff requires extending the table or using an independently derived cumulative calculation.

Reliability of a success probability: This distribution assumes p is constant and attempts are independent. If learning improves success rates after each failure, or incidents occur in correlated bursts, the simple negative-binomial model may not describe the process. Similarly, a customer-retention experiment whose trial outcomes change after interventions should not be treated as repeated identical p without data support.

• #NUM!: Zoho lists p outside 0–1, x negative and r below 1. Repair incorrect probability units such as entering 40 to mean 40%; enter 0.4 or a properly formatted 40% instead.
• Zero or one success probability: Mathematically p=0 prevents reaching positive r successes, and p=1 implies zero failures. Zoho’s documentation does not say exactly how its native implementation treats these extremes; avoid claiming a specific error/result absent execution.
• Fractions in x/r: The model counts whole failures and successes. Microsoft’s Excel docs specify integer truncation, but Zoho does not; don’t transfer Excel’s coercion rule. Use verified integers or label native behaviour untested.
• #VALUE!, #NAME!, #REF!: Zoho’s general entries concern invalid argument types, bad function names and broken references. In practice inspect the three input columns, quoted numbers and copied references before changing the model.
• Formula appears smaller than expected: Check whether you really need an exact probability or a cumulative probability (sum all x≤threshold). Also note that probabilities can become extremely small far from the likely failure counts.
• Do not sum a partial support to 1: The negative-binomial support extends to arbitrarily large x (when 0<p<1), so a table from 0 to 6 intentionally totals less than one. The missing mass is the probability of additional failures.

Zoho documents NEGBINOMDIST(x;r;p) with three arguments and no cumulative option. Microsoft documents Excel’s modern NEGBINOM.DIST(number_f,number_s,probability_s,cumulative) with a fourth argument that decides whether to return a cumulative probability, and also documents the legacy Excel NEGBINOMDIST name. Thus a four-argument modern Excel formula cannot be assumed to work unchanged in Zoho. Microsoft’s integer truncation rule is not evidence for Zoho. Zoho NEGBINOMDIST · Microsoft NEGBINOM.DIST.

For a fixed number of independent trials, compare Zoho BINOMDIST. For counts of arrivals over a time interval with a given mean, compare POISSON. For dependent draws from a fixed set without replacement, compare HYPGEOMDIST. These are checked Zoho official Help links, not internal published tutorial URLs.

Zoho Sheet evidence: This guide covers the English-language Zoho Sheet function documented in the linked official Zoho Help article, inspected 13 October 2026 (UTC). Zoho describes Sheet as an online spreadsheet product and publishes function reference pages without always specifying a function’s first supported release, rollout cohort, plan, device, mobile app edition, numerical precision guarantees or offline calculation support. Accordingly, none of those compatibility details is asserted here. Check the target account and client before claiming parity across web, Windows, macOS, Android, iOS or iPadOS. The Help knowledge base currently lists those client sections, but their existence does not establish function-level equivalence. Zoho Function Catalogue · Zoho Product Help.

Formula delimiters: The English reference writes arguments separated by semicolons (;), as do the examples here. Zoho’s workbook settings let the author change locale, decimal separator and digit grouping, and changes are shared with collaborators. Confirm parsing with the workbook actually used; do not assume formulas copied from comma-delimited Excel examples parse unchanged. Zoho Locale Settings. Dates and workbook numeric displays can also vary by locale.

Validation classification: OFFICIAL_DOCUMENTATION_RESEARCH establishes the vendor’s published names, argument order, stated modes and documented errors; INDEPENDENT_CALCULATION establishes the mathematical expected results on this page; NATIVE_ZOHO_EXECUTION_NOT_PERFORMED means no Zoho workbook/account was used to execute or screenshot these examples. A mathematical identity is not evidence of undocumented coercion or of how a particular Zoho release rounds or displays a cell.

Is this a native Zoho Sheet formula? The function name and documented signature come from official Zoho Help. The example results are independently calculated predictions; they have not been run in a Zoho session.

Can I paste the Excel version of this formula? Compare both the function name and required argument count. The compatibility paragraph below gives an evidence-based Excel distinction, and the actual workbook’s syntax and separators must be checked.

Why does an answer display as 0 or 1? Probabilities may be close to a boundary, or the cell’s number format may be rounding away significant digits. Increase displayed decimal places, verify the data scale and use a mathematically suitable model; do not equate displayed rounding with proof of exact 0 or 1.

Where do these examples come from? The worksheet inputs are invented, openly disclosed practice datasets. The outputs come from independent SciPy calculations and mathematical formulas—not from the illustrative, potentially different, private input cells in Zoho’s official screenshots or embedded demo.

• Z1 Official Zoho Sheet documentation for this function, inspected 13 October 2026: purpose, syntax, argument descriptions, examples and vendor error entries.
• [Z2] Zoho Sheet statistical functions catalogue, inspected 13 October 2026: placement in its statistical reference; catalogue page is not a native execution log.
• [Z3] Zoho Sheet spreadsheet locale and calculation settings, inspected 13 October 2026: workbook locale and separators.
• M1 Microsoft Support documentation for the named Excel counterpart, inspected 13 October 2026: Excel-specific comparison only, not proof of Zoho behaviour.