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BINOMDIST in Apple Numbers: exact and cumulative binomial probabilities

Statistical Intermediate Apple Numbers

Calculate an exact count or cumulative probability of independent Bernoulli trials using Apple Numbers BINOMDIST.

Verification status: Official documentation referenced; example outcomes independently reasoned or arithmetically checked where stated. Not executed in Apple Numbers. How formula examples are checked

Quick answer. =BINOMDIST(3,98,0.04,TRUE) is independently expected to return 0.445507210083272: the chance of three or fewer successes in 98 independent trials with 4% success chance per trial. =BINOMDIST(3,98,0.04,FALSE) instead returns 0.201402522366024, the chance of exactly three successes. Apple’s fourth argument is required, not optional. [A1, C1]

Choose the right probability question

The binomial model describes a fixed number n of independent attempts, each having two outcomes (success or failure) and the same success probability p. If X is the number of successes, there are two distinct questions:

  • Exactly k successes: P(X=k)=C(n,k) p^k(1-p)^(n-k), the probability mass.
  • k or fewer successes: P(X≤k)=Σ[j=0..k] P(X=j), the cumulative probability.

The function-type selector determines which question Numbers computes. The formula is not appropriate if the number of attempts varies, probabilities change significantly from one attempt to the next, or outcomes are dependent. A binomial result is a probability of counts, not an average outcome. [A1, C1]

Exact syntax, order, types and defaults

=BINOMDIST(success-num, trials, prob-success, function-type)
Position Argument Apple-documented type Required Documented restriction / meaning Default
1 success-num Number value Yes 0≤success-num≤trials, requested successful count None
2 trials Number value Yes trials≥0, total attempts None
3 prob-success Number value Yes 0≤p≤1 None
4 function-type Modal value Yes TRUE or 1 = CDF; FALSE or 0 = PMF None

For meaningful binomial count calculations, use whole-number successes and trials, and ensure successes do not exceed trials. Apple’s entry describes arguments as numeric and specifies the bounds, but does not document a portable rounding rule for fractional counts: do not invent one. Use TRUE/FALSE explicitly to make the mode readable, or numeric 1/0 when matching Apple’s published examples. [A1]

Build an exact-versus-cumulative Numbers table

Create a table TrialModel with B1 Successful count k, C1 P(X=k), PMF, D1 P(X≤k), CDF, F1 Trials n, G1 Success p. Put 10 in F2 and 0.3 in G2. Enter k values 0 through 10 in B2:B12. Then enter these formulas and fill downward:

C2: =BINOMDIST(B2,$F$2,$G$2,FALSE)
D2: =BINOMDIST(B2,$F$2,$G$2,TRUE)

The $ anchors keep the model assumptions fixed while B2 varies down rows. Numbers documents this reference behavior. [A1, A3]

k Exact P(X=k) Cumulative P(X≤k)
0 0.028247524900 0.028247524900
1 0.121060821000 0.149308345900
2 0.233474440500 0.382782786400
3 0.266827932000 0.649610718400
4 0.200120949000 0.849731667400
5 0.102919345200 0.952651012600
6 0.036756909000 0.989407921600
7 0.009001692000 0.998409613600
8 0.001446700500 0.999856314100
9 0.000137781000 0.999994095100
10 0.000005904900 1.000000000000

These are independently calculated mathematical values, not screen captures or recorded Numbers results. In a valid distribution, the PMF values across all k sum to 1 and the CDF ends at 1. Check this arithmetic and that the CDF never decreases; a screenshot from a different spreadsheet app is not proof about Numbers. [C1]

Advanced examples that solve useful questions

1. Probability of at least four successes

For n=10, p=.3, at least four is the complement of at most three:

=1-BINOMDIST(3,10,0.3,TRUE)

Expected result 0.3503892816. Using BINOMDIST(4,10,0.3,TRUE) would instead answer at most four, the opposite side of the cutoff. Check inclusive wording carefully: ≥4 uses 1-CDF(3), while >4 uses 1-CDF(4). [A1, C1]

2. Probability of two through five successes, inclusive

=BINOMDIST(5,10,0.3,TRUE)-BINOMDIST(1,10,0.3,TRUE)

Expected result 0.8033426667 because P(2≤X≤5)=P(X≤5)−P(X≤1). Avoid subtracting the CDF at 2: that would exclude the case of exactly two successes. [A1, C1]

3. Cross-check a directly computed point probability

For three successes in ten trials with p=0.3, =BINOMDIST(3,10,0.3,FALSE) is expected to be 0.266827932. By elementary combinatorics, C(10,3)×0.3^3×0.7^7 has the same result; that is a mathematical check, not an Apple-proven computation path. [A1, C1]

4. Turn a binomial percentile into a count threshold

=CRITBINOM(10,0.3,0.95) independently gives 5, because P(X≤4)=0.8497316674 is below .95, while P(X≤5)=0.9526510126 reaches .95. This is a count cutoff, not an exact probability. The CRITBINOM input order is different: trials, prob-success, alpha. [A6, C1]

5. Distinguish independent sampling from sampling without replacement

Inspect the experiment. Ten random cards dealt without replacement from a small deck typically lack independent, constant-p draws. Do not automatically apply BINOMDIST to such a model. A tiny sampling fraction might justify a carefully argued approximation, but Numbers’ BINOMDIST documentation does not certify a hypergeometric alternative. [A1, C1]

Troubleshooting and boundary cases

Formula / situation Interpretation Correction
=BINOMDIST(3,98,0.04) Required function-type omitted. Add TRUE for ≤3 or FALSE for exactly 3.
=BINOMDIST(11,10,0.3,TRUE) success-num>trials violates bounds. Recheck k and n.
=BINOMDIST(-1,10,0.3,FALSE) Negative success count invalid. Use a valid whole nonnegative count.
=BINOMDIST(3,10,1.2,FALSE) Probability outside [0,1]. Store p as a fraction between 0 and 1.
=BINOMDIST(3,10,30,FALSE) Entered 30, not 30%. Use 0.3 or properly formatted 30%.
Getting .64961 instead of .26683 for k=3 Chose CDF instead of PMF. Change final argument to FALSE.
k=0 or p=0/p=1 Allowed by numeric bounds, but probabilities may collapse to zero or one. Examine the degenerate model before treating it as an error.
Trials or counts entered as 3.5 Documentation lacks a rounding/coercion guarantee. Use meaningful whole-number trial counts.

Apple’s article does not assign an error token for each invalid input; don’t claim all such entries return a particular Excel error such as #NUM!. Very tiny probabilities can also be affected by finite numerical precision; avoid drawing conclusions from display rounding. [A1, A2, C1]

Apple Numbers compatibility and locale

The current English 15.4 Apple help page explicitly supports BINOMDIST, TRUE/FALSE modes and the four required positional arguments. That proves current documented syntax, not availability in every old Mac/iPhone/iPad release and not exact parity with Microsoft Excel, Google Sheets or OpenOffice. Apple release notes were separately checked; this name was not individually established as a function newly introduced in 15.4. [A1, A4, A5]

For English decimal-point syntax: =BINOMDIST(3,10,0.3,TRUE). If the locale uses comma decimals, Apple’s overview directs you to use semicolons as argument separators, such as =BINOMDIST(3;10;0,3;TRUE). Only claim syntax and typing behavior Apple actually documents; names are case-insensitive, but localized names cannot be inferred from another app. [A2]

Frequently asked questions

Does TRUE mean at least k successes? No. TRUE means k or fewer successes. Use 1-BINOMDIST(k-1,n,p,TRUE) for at least k, with attention to edge cases.

When should I use CRITBINOM? To find a count threshold, not the probability at an already known count.

Does BINOMDIST model a varying success probability? No. Apple specifically requires the binomial model’s constant probability and independent trials.

Official references and validation record

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.