Binomial distribution

The formula

The probability of getting exactly k successes in n independent trials, each with success probability p:

P(X = k) = C(n, k) · p^k · (1 − p)^(n − k)

Read it in three pieces. C(n, k) = n! / (k!·(n−k)!) counts the number of distinct ways k successes can be arranged among n trials. p^k is the probability of the k successes happening. (1 − p)^(n − k) is the probability of the other n − k trials failing. Multiply them and you have the chance of that exact count.

The two numbers you actually carry around:

mean:      μ = n·p
variance:  σ² = n·p·(1 − p)
std dev:   σ = √( n·p·(1 − p) )

Mean reads as "on average, this fraction of the trials succeed." The variance is largest at p = 0.5 and shrinks toward zero as p heads to 0 or 1 — a nearly-certain process barely scatters.

The most likely single count (the mode) is:

mode = floor( (n + 1)·p )

A quick check that the whole thing is self-consistent: sum P(X = k) over k = 0 to n and you get exactly 1. It's the binomial expansion of (p + (1−p))^n = 1^n = 1, which is where the distribution gets its name.

Where you meet it

On the receiving dock, running acceptance sampling. A lot arrives, you pull a sample of n units, and you accept the lot if the number of defectives is at or below some limit. Say you sample n = 20 and the true defect rate is p = 0.05. The probability of finding zero defectives — and accepting the lot — is (0.95)^20 = 0.358. So a plan that rejects on the first defect still waves through a 5%-defective lot 36% of the time. That curve of "probability of accepting versus true defect rate" is the operating characteristic (OC) curve, and it's binomial arithmetic under every MIL-STD / ANSI-ASQ sampling table.
At the test stand, scoring a one-shot device run. Igniters, squibs, pyrotechnic valves, airbag inflators — things you can't test twice on the same unit. Fire n = 10 off the line, get 10 successes, and someone asks "so what's the reliability?" If the true success probability were p = 0.90, the chance of a clean 10-for-10 is (0.90)^10 = 0.349. Ten successes is entirely consistent with a 90%-reliable device, which is why a run of clean shots does not by itself prove three-nines reliability. You need the binomial to turn the count into an honest confidence bound.
At the bench, tallying bit errors. Push n bits through a link with bit error ratio p, and the number of bit errors is binomial. At BER = 1e−6 over a million bits, expected errors are μ = n·p = 1, and the chance of a perfectly clean million-bit block is (1 − 1e−6)^1000000 = 0.368. That "expect about one error, but a clean block is common" intuition is the binomial collapsing into its Poisson limit.
At the review board, when someone reports "8 of 40 passed." Any time the data is a count of successes out of a fixed number of independent tries — first-pass yield, go/no-go inspection, hit rate on a tracking test — the binomial is the model that tells you whether 8/40 is meaningfully different from a required 25%.

How it works

The binomial is the sum of n Bernoulli trials — a Bernoulli trial being the atom: one event, two outcomes, probability p of the one you're counting. Add up n of them and you count successes. That decomposition is the whole model, and it's also where every gotcha lives, because it demands three things that reality doesn't always supply.

Independence is the load-bearing assumption. The trials have to not influence each other. Sample 20 parts from a tray that all came off the same drifting machine in the same five minutes and their defects are correlated, not independent — the binomial will understate your true scatter and hand you a falsely tight bound. The classic version: sampling without replacement from a small lot. Pull 20 from a batch of 50 and each draw changes the odds for the next; that's the hypergeometric distribution, not the binomial. The binomial is the with-replacement, or large-population, idealization. Rule of thumb: if the sample is under ~10% of the lot, the binomial is close enough; above that, use the hypergeometric.

Constant p is the other load-bearing assumption. Every trial needs the same success probability. If p drifts across the run — tool wear, warming test stand, a shift change — you're averaging over a moving target and the real distribution is wider than binomial. This is the pass/fail cousin of the same drift problem that fattens the tails of any statistical model built on production data.

The two large-n shortcuts, and when each is legal. When n gets big, the factorials get ugly and you reach for an approximation:

Normal approximation. When n·p and n·(1−p) are both at least about 5 (some shops insist on 9 or 10), the binomial is close to a normal with the same μ and σ. But it's a continuous curve standing in for a discrete count, so you owe it a continuity correction — add or subtract 0.5 before you convert to a z-score. Concrete: for n = 100, p = 0.5, the exact P(X ≤ 45) is 0.1841. The normal approximation with the ±0.5 correction gives 0.1841. Skip the correction and you get 0.1587 — a 14% error from one forgotten half. People drop the continuity correction constantly and then wonder why their tail probabilities are off.
Poisson approximation. When n is large and p is small (rule of thumb: n ≥ 20, p ≤ 0.05, n·p ≤ 1, though it stays useful up to n·p ≈ 10), the binomial collapses to a Poisson with λ = n·p. This is the rare-event regime — defects per wafer, errors per block, particle counts. For n = 1000, p = 0.001, exact P(X = 0) = 0.36770; Poisson with λ = 1 gives e^(−1) = 0.36788. Close enough that nobody computes the binomial.

The mistake that recurs: treating a small run as proof. Zero defects in 20 pulls does not mean the lot is clean, and 10-for-10 does not mean 100% reliable. With p = 0.90, a perfect 10-for-10 happens over a third of the time. To turn a clean count into a reliability floor you invert the binomial: the "rule of three" says that after n trials with zero failures, the upper 95% bound on the failure probability is about 3/n — so 10 clean shots buys you a failure-rate bound near 30%, not zero. That gap between "we saw no failures" and "we've proven low failure rate" is where optimistic reliability claims get made and later regretted.

History

The binomial is old enough that the counting came before the theory. The coefficients C(n, k) are the rows of Pascal's triangle, tabulated centuries before anyone spoke of probability, and Isaac Newton's generalized binomial theorem (1660s) gave the algebra of (a + b)^n its modern form. But the distribution — the idea that a count of successes has a predictable spread you can reason about — belongs to Jacob Bernoulli.

Bernoulli (1655–1705), of the Basel mathematical dynasty, did most of his work on the "mathematics of uncertainty" between roughly 1684 and 1690 ^[1]^[3]. He wrote it into a treatise called Ars Conjectandi — "The Art of Conjecturing" — which was still unfinished when he died in 1705. His nephew Nicolaus edited it and published it in Basel in 1713, eight years after Jacob's death ^[1]^[2]^[3]. In it Bernoulli worked out the distribution of successes in repeated equal-odds trials and, crucially, proved the first version of the law of large numbers: run enough trials and the observed success fraction converges on the true p, with a bound on how many trials you need to be confident ^[2]^[3]. He'd reportedly chewed on that result for twenty years before he was satisfied with it ^[3]. That theorem is the entire justification for estimating a defect rate by sampling — it's why counting works.

The trials themselves got Bernoulli's name attached much later. The individual two-outcome event is a "Bernoulli trial," but that English term is a twentieth-century coinage — its first recorded uses land in the early 1950s, roughly 250 years after the man's death ^[4]^[5]. The normal approximation to the binomial came from the other direction: Abraham de Moivre, working the binomial coefficients for large numbers of coin flips, published the first version of what became the de Moivre–Laplace theorem in the 1738 edition of The Doctrine of Chances ^[4]^[6] — the same seed that grew into the normal distribution.

Related tools

/tools/adc-resolution — bit-error tallies over a sampled channel are binomial counts
/tools/db-converter — BER and link-margin work sits next to the pass/fail counting the binomial models

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →