The Binomial Distribution | 二项分布

📚 The Binomial Distribution | 二项分布

The binomial distribution is a fundamental probability model in A‑Level Statistics. It describes the number of successes in a fixed number of independent trials, each with the same probability of success. Whether you are modelling defective items in a production line or correctly answered multiple‑choice questions, the binomial distribution gives a clear toolkit for making predictions and testing hypotheses.

二项分布是 A‑Level 统计中基础的概率模型。它描述的是固定次数的独立试验中成功的次数，每次试验成功的概率相同。无论是模拟生产线上的次品数，还是答对选择题的数量，二项分布都提供了一套清晰的工具来进行预测和假设检验。

1. What Is a Binomial Distribution? | 什么是二项分布？

A binomial distribution arises when we count the number of successes in n independent trials of a Bernoulli process. Each trial has exactly two possible outcomes – success or failure – and the probability of success, p, is constant. If the random variable X represents the number of successes, we write X ~ B(n, p).

当我们计算一个伯努利过程在 n 次独立试验中成功的次数时，就产生了二项分布。每次试验只有两种可能的结果——成功或失败——且成功的概率 p 保持不变。若随机变量 X 表示成功的次数，我们记作 X ~ B(n, p)。

For example, tossing a fair coin 10 times and recording the number of heads gives X ~ B(10, 0.5). Flipping a biased coin with a 30% chance of heads 8 times yields X ~ B(8, 0.3). The key is that the trials are identical and independent.

例如，抛一枚公平硬币 10 次并记录正面次数，则 X ~ B(10, 0.5)。抛一枚正面概率为 30% 的偏斜硬币 8 次，则 X ~ B(8, 0.3)。关键在于各次试验是完全相同且独立的。

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

To use the binomial distribution correctly, four conditions must be satisfied:

要正确使用二项分布，必须满足四个条件：

Fixed number of trials (n): The experiment consists of n repeated trials.
英文后紧随中文：固定试验次数 (n)：实验包含 n 次重复试验。
Two possible outcomes per trial: Each trial results in either ‘success’ or ‘failure’.
每次试验仅有两个结果：每次试验要么“成功”，要么“失败”。
Constant probability of success (p): The probability p remains the same for every trial.
恒定的成功概率 (p)：每次试验的 p 值保持不变。
Independence: The outcome of any trial does not affect the outcome of another.
独立性：任何一次试验的结果不影响其他试验的结果。

If any of these conditions is violated, the binomial model is inappropriate. For instance, if we draw balls from a bag without replacement, the trials become dependent, and we need the hypergeometric distribution.

如果其中任何一个条件被违反，二项模型就不再适用。例如，在不放回地从袋中抽取球时，试验之间变得不独立，此时需要使用超几何分布。

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

The probability of obtaining exactly r successes in n trials is given by:

在 n 次试验中恰好得到 r 次成功的概率由下式给出：

P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ

where C(n, r) is the binomial coefficient, also written as nCr or ⁿCᵣ, and equals n! / (r! (n − r)!). The term (1 − p) is often called q, so the formula becomes P(X = r) = C(n, r) pʳ qⁿ⁻ʳ.

其中 C(n, r) 是二项式系数，也写作 nCr 或 ⁿCᵣ，等于 n! / (r! (n − r)!)。(1 − p) 常记作 q，因此公式也可写为 P(X = r) = C(n, r) pʳ qⁿ⁻ʳ。

For a calculator, the combination function is usually labelled nCr. You must be comfortable using it to compute coefficients by hand or with technology, as formula questions are common in Edexcel exams.

在计算器上，组合函数通常标注为 nCr。你必须能熟练地手工或借助计算器计算组合系数，因为这类公式题在 Edexcel 考试中很常见。

4. Worked Example – Calculating Individual Probabilities | 计算示例 – 单个概率

Suppose a fair die is rolled 5 times. Success is defined as rolling a six, so p = 1/6, n = 5. Find the probability of getting exactly two sixes.

假设掷一颗公平骰子 5 次。定义掷出 6 点为成功，因此 p = 1/6，n = 5。求恰好得到两个 6 的概率。

Using the formula: P(X = 2) = C(5, 2) (1/6)² (5/6)³.

使用公式：P(X = 2) = C(5, 2) (1/6)² (5/6)³。

C(5, 2) = 10. (1/6)² = 1/36. (5/6)³ = 125/216. Multiply: 10 × 1/36 × 125/216 = 1250 / 7776 ≈ 0.1608.

C(5, 2) = 10。(1/6)² = 1/36。(5/6)³ = 125/216。相乘：10 × 1/36 × 125/216 = 1250 / 7776 ≈ 0.1608。

So there is about a 16.1% chance of rolling exactly two sixes in 5 rolls.

因此，在 5 次投掷中恰好出现两个 6 的概率约为 16.1%。

For finding multiple probabilities, building a probability distribution table can be useful. Below is the full distribution for B(5, 1/6):

若需求出多个概率，构建概率分布表会很有用。以下是 B(5, 1/6) 的完整分布：

r	P(X = r)
0	(5/6)⁵ ≈ 0.4019
1	5×(1/6)×(5/6)⁴ ≈ 0.4019
2	10×(1/6)²×(5/6)³ ≈ 0.1608
3	10×(1/6)³×(5/6)² ≈ 0.0322
4	5×(1/6)⁴×(5/6) ≈ 0.0032
5	(1/6)⁵ ≈ 0.0001

The probabilities sum to 1, which provides a quick check of your calculations.

所有概率之和为 1，这可以用来快速检查你的计算。

5. Cumulative Binomial Probabilities | 累积二项概率

Often we need P(X ≤ k) or P(X ≥ k) rather than exact values. Cumulative probabilities can be found by summing individual probabilities or by using the binomial cumulative distribution tables provided in the Edexcel formula booklet.

我们通常需要 P(X ≤ k) 或 P(X ≥ k)，而不是单一取值概率。累积概率可以通过逐项求和得到，或利用 Edexcel 公式手册中提供的二项累积分布表获得。

For example, if X ~ B(10, 0.5), to find P(X ≤ 3) we sum P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3). Using a table, we read the value directly: P(X ≤ 3) = 0.1719 (approx).

例如，若 X ~ B(10, 0.5)，求 P(X ≤ 3)，我们将 P(X=0)+P(X=1)+P(X=2)+P(X=3) 相加。查表可直接得到 P(X ≤ 3) = 0.1719（近似值）。

To find P(X ≥ 7), we can use the complement rule: P(X ≥ 7) = 1 − P(X ≤ 6). If P(X ≤ 6) = 0.8281, then P(X ≥ 7) = 0.1719. This symmetry arises because p = 0.5.

求 P(X ≥ 7) 时，可以使用补集法则：P(X ≥ 7) = 1 − P(X ≤ 6)。若 P(X ≤ 6) = 0.8281，则 P(X ≥ 7) = 0.1719。由于 p = 0.5，这里出现了对称性。

Be careful with strict inequalities: P(X < 5) = P(X ≤ 4), and P(X > 2) = 1 − P(X ≤ 2). These conversions are tested frequently.

务必小心严格不等式：P(X < 5) = P(X ≤ 4)，P(X > 2) = 1 − P(X ≤ 2)。这类转换经常出现在考题中。

6. Using the Binomial Cumulative Tables | 使用二项累积表

Edexcel provides cumulative binomial tables for various n and p values (typically p ≤ 0.5). When p > 0.5, you can swap the definition of success and failure to use the table. For instance, if X ~ B(10, 0.8) and you need P(X ≤ 7), let Y be the number of failures, Y ~ B(10, 0.2). Then P(X ≤ 7) = P(Y ≥ 3) = 1 − P(Y ≤ 2). This technique saves time.

Edexcel 提供了不同 n 和 p（通常 p ≤ 0.5）的二项累积表。当 p > 0.5 时，你可以对调成功和失败的定义来使用表格。例如，X ~ B(10, 0.8)，需要求 P(X ≤ 7)，令 Y 为失败次数，Y ~ B(10, 0.2)。则 P(X ≤ 7) = P(Y ≥ 3) = 1 − P(Y ≤ 2)。这一技巧能节省时间。

Tables often list P(X ≤ r) for r = 0, 1, …, n. A small extract for B(5, 0.5) looks like this:

表格通常列出 r = 0, 1, …, n 时的 P(X ≤ r)。下面对 B(5, 0.5) 的简表如下：

r	P(X ≤ r)
0	0.03125
1	0.1875
2	0.5
3	0.8125
4	0.96875
5	1

Always double‑check which table you are reading – individual or cumulative – as misreading can cost marks.

务必确认你读的是哪类表——单项还是累积——因为读错表可能导致失分。

7. Mean and Variance of a Binomial Distribution | 二项分布的均值和方差

If X ~ B(n, p), the expected number of successes (mean) and the variance are:

若 X ~ B(n, p)，成功的期望值（均值）和方差为：

E(X) = np

Var(X) = np(1 − p)

The standard deviation is their square root: SD(X) = √[np(1 − p)]. These formulas are derived from the fact that X is the sum of n independent Bernoulli(p) variables.

标准差为其平方根：SD(X) = √[np(1 − p)]。这些公式源于 X 是 n 个独立 Bernoulli(p) 变量之和这一事实。

For example, in 100 rolls of a fair die, the number of sixes has mean = 100 × 1/6 ≈ 16.67 and variance = 100 × 1/6 × 5/6 ≈ 13.89. These summaries are vital for making predictions and for normal approximations later.

例如，在 100 次公平骰子投掷中，出现 6 点的次数均值为 100 × 1/6 ≈ 16.67，方差为 100 × 1/6 × 5/6 ≈ 13.89。这些摘要量对于预测和今后的正态近似至关重要。

Exam questions often ask you to verify that a given scenario satisfies conditions, then to calculate the mean and variance, and finally to interpret them in context.

试题常要求你先验证给定情境是否满足条件，然后计算均值和方差，最后在实际背景下加以解释。

8. Hypothesis Testing with the Binomial Distribution | 二项分布假设检验

One of the most important applications is one‑tailed and two‑tailed hypothesis tests for a proportion. The test statistic is X ~ B(n, p₀), where p₀ is the hypothesised probability of success under the null hypothesis H₀.

二项分布最重要的应用之一是对比率进行单尾和双尾假设检验。检验统计量为 X ~ B(n, p₀)，其中 p₀ 是零假设 H₀ 下的假设成功概率。

Steps for a hypothesis test:

假设检验的步骤：

State hypotheses: H₀: p = p₀, H₁: p < p₀ (left‑tailed), p > p₀ (right‑tailed) or p ≠ p₀ (two‑tailed).
陈述假设：H₀: p = p₀，H₁: p < p₀（左尾），p > p₀（右尾）或 p ≠ p₀（双尾）。
Specify test statistic and significance level α: e.g. X ~ B(20, 0.5), α = 0.05.
明确检验统计量和显著性水平 α：例如 X ~ B(20, 0.5)，α = 0.05。
Find the probability of the observed result or more extreme: p‑value = P(X ≤ observed) for left tail, etc.
计算观测结果或更极端情形的概率：左尾 p‑值 = P(X ≤ 观测值) 等。
Compare p‑value with α or compare test statistic with critical region: If p‑value < α, reject H₀.
将 p‑值与 α 比较，或将检验统计量与临界区域比较：若 p‑值 < α，拒绝 H₀。
Conclude in context: e.g. “There is sufficient evidence at the 5% level to reject the claim that p = 0.5.”
结合背景得出结论：例如，“在 5% 水平下有充分证据拒绝 p = 0.5 的声称。”

Worked example: A company claims 70% of its seeds germinate. A gardener plants 20 seeds and only 10 germinate. Test at the 5% significance level whether the germination rate is less than 70%.

解析示例：某公司声称其种子的发芽率为 70%。一位园丁种下 20 颗种子，只有 10 颗发芽。在 5% 显著性水平下检验发芽率是否低于 70%。

H₀: p = 0.7, H₁: p < 0.7. X ~ B(20, 0.7). Observed successes = 10. p‑value = P(X ≤ 10). Using tables: P(X ≤ 10) = 0.0480 (approx). Since 0.0480 < 0.05, reject H₀. There is sufficient evidence that the true germination rate is below 70%.

H₀: p = 0.7，H₁: p < 0.7。X ~ B(20, 0.7)。观测成功次数 = 10。p‑值 = P(X ≤ 10)。查表得 P(X ≤ 10) = 0.0480（近似）。由于 0.0480 < 0.05，拒绝 H₀。有充分证据表明实际发芽率低于 70%。

In a two‑tailed test, you halve the significance level for each tail or double the one‑tailed p‑value before comparing. Edexcel expects you to state the critical region clearly, e.g. X ≤ c₁ or X ≥ c₂.

在双尾检验中，你需要将显著性水平平分到两个尾部，或在比较前将单尾 p‑值加倍。Edexcel 要求你清晰地陈述临界区域，例如 X ≤ c₁ 或 X ≥ c₂。

9. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧

Misidentifying ‘success’: Define success clearly – it does not have to be the desirable outcome. In exam questions, read carefully what is being counted.

错误定义“成功”：务必清楚定义成功——成功未必指称心如意结果。在考试中，仔细阅读题目计数对象。

Independence assumption: Check that trials are independent. If sampling without replacement and sample size is less than 10% of population, binomial can still be used as an approximation, but state your reasoning.

独立性假设：检查试验是否独立。如果是不放回抽样且样本量小于总体的 10%，二项分布仍可作为近似，但需陈述理由。

Table navigation errors: When p > 0.5, remember to invert successes and failures. Practise with the formula booklet so you can find values quickly.

查表错误：当 p > 0.5 时，记得交换成功与失败的定义。多练习使用公式手册，以便迅速查找数值。

Inequality handling: Convert P(X > 5) to 1 − P(X ≤ 5) correctly. This is one of the most common slip‑ups. Write out the conversion before picking up the calculator.

不等式处理：正确地将 P(X > 5) 转换为 1 − P(X ≤ 5)。这是最常出现的疏漏之一。在拿起计算器前，先写出转换式。

Hypothesis test conclusions: Always phrase your conclusion in the context of the problem, using ‘sufficient evidence’ or ‘not enough evidence’, and mention the significance level. Never say ‘prove’ – hypothesis tests provide evidence, not proof.

假设检验结论：始终结合问题背景撰写结论，使用“充分证据”或“证据不足”，并提及显著性水平。切勿使用“证明”——假设检验提供的是证据，而非证明。

Using symmetrical properties: For p = 0.5, P(X ≤ r) and P(X ≥ n − r) are equal, which can speed up calculations. Recognise symmetry to save time.

利用对称性：当 p = 0.5 时，P(X ≤ r) 与 P(X ≥ n − r) 相等，这可以加速计算。识别对称性以节省时间。

Combining these strategies will help you tackle binomial questions confidently and accurately in the Edexcel exam.

结合这些策略，你将能在 Edexcel 考试中自信而准确地解答二项分布的题目。

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The Binomial Distribution | 二项分布