BINOMDIST

What it does

BINOMDIST(number_s, trials, probability_s, cumulative)

BINOMDIST(number_s, trials, probability_s, cumulative) describes successes in a fixed number of independent trials with one success probability. FALSE returns exactly that many successes; TRUE returns at most that many. BINOMDIST is the compatibility name; BINOM.DIST makes the distribution role explicit.

Try it

=BINOMDIST(3,3,0.5,FALSE) is about 0.125. =BINOMDIST(3,3,0.5,TRUE) returns 1, since all possible outcomes have at most three successes.

Quirks

Counts are truncated toward zero. A probability outside zero to one, or an out-of-range success count, gives #NUM!; nonnumeric numeric arguments give #VALUE!. “At least k” uses a complement at k-1, not at k. The all-success mass and its cumulative endpoint follow different paths.

How to implement

After truncating successes k and trials n toward zero, validate 0<=k<=n and 0<=p<=1. The current recorded domain rejects n>=2147483647 before selecting endpoint shortcuts. At p=0 or p=1, handle the degenerate distribution explicitly; the recorded noncumulative p=1,k=0,n>0 case gives #NUM!, a compatibility exception to the mathematical zero.

For the all-success, noncumulative route k=n>0, 0<p<1:

1. Obtain the stored measured public LN result for p, called L.
2. Compute t=R(n*L) with the stored-arithmetic recipe, rounding once to a 64-bit significand and then storing to binary64.
3. Pass the stored t to the measured public EXP route and publish its result under the function’s finite/subnormal policy.

Worked n=3,p=0.5: L is about -0.6931471805599453, the stored product is about -2.0794415416798357, and this reconstruction’s exponential is 0.12500000000000003, near the mathematical probability 0.125. The stored intermediate values explain why the computation need not return the exactly representable mathematical result. A direct pow(p,n) has different intermediate stages and is not a substitute for this identified computation tree.

The all-success stored product is distinguished by 300 numeric discriminants and 256 controls in the cited code. Its scope does not cover the zero-success producer, interior masses or every separately recorded domain answer.

The later R31 source isolates additional interior-mass stages. These start from Loader’s published saddle-point form, then preserve separately measured arithmetic sites. Let D mean ordinary binary64 storage and E resident 64-bit-significand arithmetic; their portable definitions keep E distinct from R.

1. For 0<k<n, set q=D(1-p). Obtain Stirling correction s and deviance d from the explicit Poisson helper recipes.
2. Form c=R(s(n)-s(k)), then successively c=R(c-d(k,R(n*p))), c=R(c-s(n-k)) and c=R(c-d(n-k,R(n*q))). Keep this subtraction order. The source records 13,000 controls across six (n,k) pairs for that exponent grid.
3. Form the ordered binary64 terms [-LN(n)/2,LN(k)/2,LN(n-k)/2,0.9189385332046728]. Their mathematical effect supplies sqrt(n/(2*pi*k*(n-k))) without directly forming that ratio.
4. Initialize stored a and resident b from c. For each term, update a=R(a-term) and separately b=E(b-term), retaining b across the whole list. Compute lost=D(E(b-a)), factor=R(1+lost), and return R(EXP(a)*factor).

BINOM uses one correction for the running resident difference. POISSON’s prefactor instead keeps one correction factor for each of its two subtractions. They are distinct reconstructions even though both compensate a stored exponent.

For k=3,n=10,p=0.25, the stated interior reconstruction gives 0.2502822875976563. Direct exact rational evaluation of 120*(1/4)^3*(3/4)^7 gives the representable mathematical value 0.25028228759765625. The first number is a worked reconstruction, rather than a new Excel reading or a statement that the candidate is correct on that input.

The current cumulative route sums masses forward with ordinary binary64 additions and caps the final sum at one; it has its own endpoint shortcuts. A stable recurrence or incomplete-beta CDF is another mathematical algorithm with a different rounding history. The source’s zero-success logarithm remains unidentified, the Stirling table and deviance retain exceptions, and the problem report still records unresolved distribution answers. These stages do not establish whole-function agreement; the separately pinned interactive calculator can also predate the later interior update.

Rust code

71 binomdist "BINOMDIST" (4, 4) [V "number_s", V "trials", V "probability_s", V "cumulative"] |s, a, sv| {
    let [n, t, p] = nums::<3>(s, a)?;
    out(s, dd::binom_dist_kernel(n, t, p, flag(s, a, 3)?))
}

No implementation has been published with a Handbook test suite for this function.

Version differences

No same-input comparison between named Excel builds is recorded for this function here. The documented behaviour below is not a measurement of every Excel release.

Alternatives

BINOM.INV finds a success-count quantile; BINOM.DIST.RANGE covers an interval of counts; HYPGEOM.DIST models sampling without replacement.

Named variants

No named variants are recorded yet.