故事 · 把一次成败重复 n 遍
Origin Story · One Trial, Repeated n Times
十七世纪末，瑞士数学家 Jacob Bernoulli 把"一次成功或失败"的最简单随机试验称作伯努利试验。
把 n 次互相独立、成功概率都是 p 的伯努利试验加起来，问"总共成功几次"，答案就是二项分布。
公式里的 C(n, k) 是组合数，回答"n 个位置里挑 k 个放成功，有多少种排法"；后面的 pᵏ(1−p)ⁿ⁻ᵏ
是每一种排法发生的概率。它最早服务于赌局与保险：n 局里赢几局、n 个保单出几个险。
今天它是抽样检验、不良品计数、A/B 测试的基石 —— 凡是数"计数型的成败"，背后都是它。
In the late 17th century, the Swiss mathematician Jacob Bernoulli labeled the simplest random experiment — one trial, success or fail — a Bernoulli trial.
Stack n independent Bernoulli trials, each with the same success probability p, ask "how many successes in total," and the answer is the binomial distribution.
The C(n, k) in the formula is the binomial coefficient, counting the ways to pick which k of the n slots are the successes; the pᵏ(1−p)ⁿ⁻ᵏ that follows is the probability of any one such arrangement.
It was first put to work on gambling and insurance: wins out of n hands, claims out of n policies.
Today it is the bedrock of acceptance sampling, defect counting, and A/B testing — any time you are counting wins and losses, this is what's underneath.
1 p 拉偏它、n 拉大它，柱子悄悄长成钟形 Skew it with p, stretch it with n — the bars grow into a bell
对称2 方差 np(1−p)：p=0.5 时最不确定 Variance np(1−p): p = 0.5 is peak uncertainty
固定 n，方差随 p 走出一条抛物线：p 越靠近 0.5，结果越难猜、波动越大；p 越靠近 0 或 1，几乎必然，波动趋零。 Hold n fixed and variance traces a parabola in p: the closer p sits to 0.5, the harder the outcome is to call and the wider it swings; push p toward 0 or 1 and the result is nearly certain, the spread collapses to zero.
3 现实里的二项分布 The Binomial in the Real World
不良品计数：抽 n 件查出几件不良，是来料检验、批次放行判定的基本模型。
Defect counting: how many defects show up in a sample of n — the basic model behind incoming inspection and lot release decisions.
抽样检验：AQL 抽样计划用二项算"抽 n 件接受或拒收"的概率，制定 OC 曲线。
Acceptance sampling: AQL plans use the binomial to compute the accept/reject probability for a sample of n, which is exactly what the OC curve plots.
A/B 测试：n 个访客里几个转化，转化次数服从二项，是显著性检验的输入。
A/B testing: how many of n visitors convert. The conversion count is binomial — the raw input for any significance test.
质量抽检：批次合格判定、单次成功次数统计，凡是"数成败"的属性数据都用它。
Quality inspection: lot conformance calls, attribute counts of any kind — anywhere you are tallying pass/fail data, the binomial is the model.
一句话In One Line
二项分布回答的是离散世界里最常见的一问：n 次独立成败里，成功几次。
它由两个参数定形状：p 控制偏斜 —— 越靠 0.5 越对称，越偏两端越歪；n 控制平滑 ——
n 一大，离散的柱群就趋近连续的正态钟形（这正是下一页"棣莫弗-拉普拉斯定理"要讲的来历）。
它还和泊松、正态是亲戚：n 大、p 小时趋近泊松，n 大、p 适中时趋近正态。
掌握了它，离散计数的统计世界就有了第一块基石。
The binomial answers the most common question in the discrete world: how many successes in n independent trials?
Two parameters fix its shape — p controls skew (closer to 0.5 is more symmetric, pushed to the edges it leans), and n controls smoothness:
push n high enough and the discrete bars converge on the continuous normal bell (exactly what the next page, the De Moivre–Laplace theorem, explains).
It also has close relatives — large n, small p drifts toward Poisson; large n with moderate p drifts toward normal.
Master it and you have the first cornerstone of the entire discrete-counting world.
常见误用Common Mistakes
试验之间并不独立还硬套二项。连续生产、有传染或批次相关时独立假设失效，结果会偏。
Forcing the binomial onto trials that aren't independent. Continuous production, contagion effects, or batch-level correlation break the independence assumption — your answer will be biased.
从有限批次不放回抽样用二项。不放回抽样应用超几何分布；只有批量远大于样本时才近似二项。
Using the binomial for sampling without replacement from a finite lot. Sampling without replacement is hypergeometric; the binomial is only a good approximation when the lot is far larger than the sample.
p 在过程中会漂移却当常数。二项要求每次 p 相同；若过程不稳定，先用控制图确认 p 稳定。
Treating p as constant when it drifts through the process. The binomial demands the same p on every trial; if the process is unstable, prove p is stationary with a control chart first.