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The Binomial Distribution: Conditions, Formula, Mean and Variance | 二项分布：条件、公式、均值与方差

2026-08-29 20:15 · 本文部分内容由AI辅助创作

📚 The Binomial Distribution: Conditions, Formula, Mean and Variance | 二项分布：条件、公式、均值与方差

The binomial distribution is one of the most important discrete probability models in the Edexcel A Level Mathematics specification. It counts the number of successes in a fixed number of independent trials, where each trial has the same probability of success. This article covers its conditions, probability formula, mean and variance, cumulative probabilities, calculator use, modelling decisions and the common exam techniques that turn a shaky answer into full marks.

二项分布是爱德思 A Level 数学考试大纲中最重要的离散概率模型之一。它统计在固定次数的独立试验中成功的次数，并且每次试验的成功概率相同。本文将介绍它的使用条件、概率公式、均值与方差、累积概率、计算器操作、建模判断，以及能把"差一点"的答案变成满分的常见考试技巧。

1. What is the Binomial Distribution? | 什么是二项分布？

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials. 'Success' does not have to mean a good outcome; it is simply the outcome we are counting. If we count the number of red cars, the number of broken bulbs or the number of people who catch a cold, the 'success' is just whatever we decided to count.

二项分布是一种离散概率分布，描述在固定次数的独立试验中成功的次数。"成功"不一定表示好的结果，它只是我们要计数的那个结果。无论是统计红色汽车的数量、损坏灯泡的数量，还是感冒人数，"成功"都只是我们事先选定要计数的那个结果。

For example, if you toss a fair coin 10 times and count the number of heads, the total number of heads follows a binomial distribution. The random variable is discrete because it can only take the whole-number values 0, 1, 2, ..., 10, and there are no possible values in between.

例如，如果你抛一枚均匀硬币 10 次并记录正面朝上的次数，正面次数就服从二项分布。这个随机变量是离散的，因为它只能取 0、1、2、…、10 这些整数值，之间不存在任何可能的取值。

Other real-life situations include the number of defective items in a batch, the number of patients who recover after a treatment, or the number of sixes rolled in 20 dice throws. In every case the structure is identical: a fixed number of repeated trials, each with the same two possible outcomes and the same probability of success.

其他实际例子包括一批产品中次品的数量、接受治疗后康复的患者人数，或者掷骰子 20 次出现 6 点的次数。在每种情形中，结构都是完全相同的：固定次数的重复试验，每次试验只有两种可能结果，且成功概率保持不变。

Because the model is built from repeated identical trials, it is sometimes called a model of 'sampling with replacement', since each trial must behave as though it knows nothing about the results of the trials before it.

由于该模型建立在重复且完全相同的试验之上，它有时被称为"有放回抽样"模型，因为每次试验都必须表现得好像完全不知道之前试验的结果。

2. Conditions for a Binomial Model | 二项模型的条件

A situation can be modelled by a binomial distribution only if all four conditions hold. Any one failure of these conditions means the binomial distribution is no longer a valid model, and examiners frequently award a mark simply for checking them.

只有同时满足以下四个条件，才能用二项分布建模。任何一个条件不成立，二项分布就不再是有效的模型，而考官经常仅仅因为考生进行了条件检验就给分。

A classic failure of the independence condition is sampling without replacement from a small population. If you draw 5 cards from a pack of 52 without replacing them, the probability of drawing a heart changes after each draw, so the binomial model does not apply directly.

独立性条件失败的典型例子是从一个小总体中进行无放回抽样。如果你从 52 张牌中不放回地抽出 5 张，那么每抽一次之后，抽到红桃的概率都会改变，因此二项模型不能直接适用。

In practice, when the sample is small compared with the population (a common rule of thumb is less than about 10% of the population), the change in probability is negligible and the binomial model is still accepted as a reasonable approximation.

在实践中，当样本相对于总体很小时（常用经验法则是样本量小于总体的约 10%），概率的变化可以忽略不计，二项模型仍被认为是合理的近似。

3. Notation, Parameters and the Range of the Variable | 记号、参数与变量的取值范围

We write X ~ B(n, p) to mean that the random variable X has a binomial distribution with n trials and probability of success p. The letter n and the letter p are called the parameters of the distribution, and between them they completely determine every probability.

我们写作 X ~ B(n, p)，表示随机变量 X 服从参数为 n 次试验、成功概率为 p 的二项分布。字母 n 与 p 称为该分布的参数，这两个参数共同完全决定了所有概率。

e is no such thing as 2.5 successes, and probabilities such as P(X = 3.4) are always zero.

变量 X 只能取从 0 到 n（含端点）的整数值。不存在"2.5 次成功"这种情况，因此像 P(X = 3.4) 这样的概率永远等于 0。

ful to define q = 1 − p, the probability of failure. The two probabilities always satisfy:

定义 q = 1 − p（失败的概率）很有用。这两个概率始终满足：

p + q = 1,   and   Σ P(X = r) = 1 for r = 0, 1, 2, …, n

That second statement simply says the probabilities of all possible outcomes add up to 1, which is a convenient check on any set of answers you produce.

第二个等式只是说明所有可能结果的概率之和为 1，这是检验你所给出一组答案是否合理的便捷方法。

Always define your variable in words before quoting the distribution. A line such as 'Let X be the number of defective components in a sample of 20' followed by 'X ~ B(20, 0.08)' earns method marks even if the arithmetic later goes wrong.

在写出分布之前，一定要用文字定义你的变量。像"设 X 为 20 个样本中次品的数量"这样的句子，后接"X ~ B(20, 0.08)"，即使后面计算出错，也能拿到方法分。

4. The Binomial Probability Formula | 二项概率公式

To find the probability of exactly r successes in n trials, we combine three ideas: choosing which trials are the successes, raising p to the power r, and raising (1 − p) to the power n − r.

要计算 n 次试验中恰好有 r 次成功的概率，我们需要综合三个想法：选出哪些试验是成功、把 p 乘方 r 次，以及把 (1 − p) 乘方 n − r 次。

P(X = r) = ⁿCᵣ × pʳ × (1 − p)ⁿ⁻ʳ

Here ⁿCᵣ, also written as C(n, r), is the number of ways of choosing which r of the n trials are the successes. It can be evaluated as:

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其中 ⁿCᵣ（也写作 C(n, r)）是从 n 次试验中选出 r 次成功的方法数。其计算方式为：

ⁿCᵣ = n! ÷ ( r! × (n − r)! )

Each individual arrangement of r successes and n − r failures has probability pʳ(1 − p)ⁿ⁻ʳ, and the factorial factor simply counts how many such arrangements exist. This is exactly the structure of the binomial expansion of (p + q)ⁿ, which is where the distribution gets its name.

每一种由 r 次成功和 n − r 次失败组成的具体排列，其概率都是 pʳ(1 − p)ⁿ⁻ʳ，而组合数只是统计这样的排列共有多少种。这正是 (p + q)ⁿ 二项式展开的结构，二项分布的名字也由此而来。

0.5 the distribution is skewed to the right, and when p is greater than 0.5 it is skewed to the left.

当 p = 0.5 时，分布关于中心完全对称，因此 P(X = r) = P(X = n − r)。当 p 小于 0.5 时，分布向右偏斜；当 p 大于 0.5 时，分布向左偏斜。

5. Worked Example: An Exact Probability | 例题：计算精确概率

Suppose a machine produces components of which 30% nspector selects 10 components at random. Let X be the number of faulty components. Then X ~ B(10, 0.3), and we want P(X = 4).

假设某机器生产的元件中有 30% 是次品，质检员随机抽取 10 个元件。设 X 为次品数量，则 X ~ B(10, 0.3)，我们要求 P(X = 4)。

Substituting into the formula with n = 10, r = 4 and p = 0.3 gives:

将 n = 10、r = 4、p = 0.3 代入公式，得到：

P(X = 4) = ¹⁰C₄ × 0.3⁴ × 0.7⁶ = 210 × 0.0081 × 0.117649 ≈ 0.2001

The probability that exactly four components are faulty is therefore about 0.200, or roughly a 20% chance. Notice that we did not round any intermediate value; rounding 0.3⁴ early would have changed the third decimal place.

因此恰好有 4 个次品的概率约为 0.200，也就是大约 20% 的机会。注意我们没有对任何中间结果提前取近似值；如果过早把 0.3⁴ 四舍五入，就会改变第三位小数。

n 3 lies close to 4 but far from 0. Checks of this kind catch sign errors and mixed-up values of p.

作为快速检验，我们可以算出 P(X = 0) = 0.7¹⁰ ≈ 0.0282，它必须小于 P(X = 4)，因为均值 3 靠近 4 而远离 0。这类检验可以捕捉符号错误或 p 值取反的问题。

If the question instead says that 70% are faulty, the model becomes X ~ B(10, 0.7) and P(X = 4) = ¹⁰C₄ × 0.7⁴ × 0.3⁶ ≈ 0.0368. The 'success' label has simply swapped sides.

如果题目改为 70% 是次品，模型就变成 X ~ B(10, 0.7)，此时 P(X = 4) = ¹⁰C₄ × 0.7⁴ × 0.3⁶ ≈ 0.0368。"成功"的标签只是换到了另一侧。

6. Mean and Variance | 均值与方差

For X ~ B(n, p) the mean and variance are remarkably simple, and you should be able to state and use them instantly, without relying on a formulae booklet.

对于 X ~ B(n, p)，其均值与方差的形式极为简洁，你应当能够立刻写出并使用它们，而不必依赖公式册。

E(X) = np    Var(X) = np(1 − p) = npq    σ = √( np(1 − p) )

is largest when p = 0.5 and shrinks towards zero as p approaches 0 or 1, because the outcome becomes more predictable.

均值的直观含义很清楚：如果重复试验 n 次，每次成功概率为 p，那么平均期望得到 np 次成功。当 p = 0.5 时方差最大；当 p 趋向 0 或 1 时方差趋近于零，因为结果变得越来越可预测。

Using the earlier example X ~ = 3 faulty components, the variance is 10 × 0.3 × 0.7 = 2.1, and the standard deviation is √2.1 ≈ 1.449.

用前面的例子 X ~ B(10, 0.3)：均值为 10 × 0.3 = 3 个次品，方差为 10 × 0.3 × 0.7 = 2.1，标准差为 √2.1 ≈ 1.449。

A common exam trap is being asked for the standard deviation when the question says 'spread' or 'typical deviation from the mean'. Remember to take the square root at the end, and keep the units of the original variable.

常见的考试陷阱是题目用"离散程度"或"偏离均值的典型幅度"来提问，实际要求的是标准差。记住最后要开平方，并保留原变量的单位。

A second trap is proportional scaling. If n doubles while p stays fixed, the mean doubles, the variance doubles, but the standard deviation is multiplied by √2 rather than by 2.

第二个陷阱是比例缩放。如果 n 加倍而 p 不变，均值加倍，方差加倍，但标准差乘以 √2 而不是 2。

7. Cumulative Probabilities and Inequalities | 累积概率与不等式

Most exam questions do not ask for exactly one value. They ask for 'at least', 'at most', 'fewer than' or 'between', and all of these are handled by converting to a cumulative probability P(X ≤ r).

大多数考题并不要求恰好某一个取值的概率，而是问"至少""至多""少于"或"介于"某范围，这些都可以通过转化为累积概率 P(X ≤ r) 来处理。