📚 Approximating a Binomial Distribution | 二项分布的近似
In A-Level Mathematics, the binomial distribution is a fundamental model for repeated independent trials with two outcomes. However, when the number of trials n becomes large, calculating exact binomial probabilities becomes tedious or even infeasible without powerful software. Fortunately, under certain conditions, we can approximate the binomial distribution using either the normal distribution or the Poisson distribution, which greatly simplifies calculations while maintaining acceptable accuracy.
在A-Level数学中，二项分布是刻画重复独立试验和两种结果的经典模型。但当试验次数 n 很大时，手动计算精确二项概率会变得极为繁琐，甚至无法完成。好在在特定条件下，我们可以用正态分布或泊松分布来近似二项分布，这样既能极大简化计算，又能保持可接受的准确度。
1. Binomial Distribution Recap and the Need for Approximation | 二项分布回顾与近似的必要性
A binomial random variable X ~ ials, each with success probability p. The probability mass function is P(X = k) = nCk × pk × (1-p)n-k luating something like P(X ≥ 200) when n=500 and p=0.4 by direct summation is impractical.
二项随机变量 X ~ B(n, p) 表示 n 次独立试验中成功的次数，每次成功概率为 p。其概率质量函数为 P(X = k) = nCk × pk × (1-p)n-k。尽管公式精确，但涉及阶乘和乘方，当 n 很大时计算极为不便。在实际考题中，若要求 P(X ≥ 200) 且 n=500, p=0.4，直接累加几乎不可行。
The two main approximations are the normal approximation (useful when p is not too extreme) and the Poisson approximation (suitable for rare events). Choosing the correct approximation is a key skill tested in Edexcel A-Level Mathematics.
两种主要的近似方法是正态近似（当 p 不太极端时适用）和泊松近似（适用于稀有事件）。选择正确的近似方法是爱德思 A-Level 数学所测试的一项关键技能。
2. Conditions for the Normal Approximation | 正态近似的条件
To approximate a binomial distribution X ~ B(n, p) with a normal distribution, we usually require that both np and n(1-p) are greater than 5 (some textbooks use 10, but Edexcel typically adopts the np > 5 and nq > 5 guideline). This ensures the binomial distribution is sufficiently symmetric and bell-shaped. The approximating normal distribution is N(np, np(1-p)).
要用正态分布近似二项分布 X ~ B(n, p)，通常要求 np 和 n(1-p) 均大于 5（部分教材用 10，但爱德思通常采用 np > 5 且 nq > 5）。这保证二项分布足够对称且呈钟形。近似正态分布为 N(np, np(1-p))。
- The larger the value of n and the closer p is to 0.5, the better the approximation.
- n 越大且 p 越接近 0.5，近似效果越好。
- If p is very close to 0 or 1, the distribution is skewed and the normal approximation becomes poor unless n is extremely large.
- 若 p 非常接近 0 或 1，分布呈偏态，除非 n 极大，否则正态近似效果很差。
In summary, check np > 5 and n(1-p) > 5 before using the normal approximation.
总之，在使用正态近似之前，应先检查 np > 5 且 n(1-p) > 5。
3. Continuity Correction Explained | 连续性修正解释
g the discrete x-value by ±0.5 to correspond to the boundaries of the continuous interval.
二项分布是离散的，而正态分布是连续的。为了用连续正态曲线近似诸如 P(X = 15) 或 P(X ≤ 20) 这样的离散概率，我们需要进行连续性修正。即通过 ±0.5 调整离散 x 值，使之对应连续区间的边界。
Here are the typical conversions:
以下是典型的转换方式：
| Discrete Probability | Normal Approximation with Continuity Correction |
|---|---|
| P(X = k) | P(k - 0.5 ≤ Y ≤ k + 0.5) |
| P(X ≤ k) | P(Y ≤ k + 0.5) |
| P(X < k) | P(Y ≤ k - 0.5) |
| P(X ≥ k) | P(Y ≥ k - 0.5) |
| P(X > k) | P(Y ≥ k + 0.5) |
这里的 Y 表示正态变量 Y ~ N(np, np(1-p))。连续性修正对改善近似至关重要，在考试中遗漏会导致失分。
4. Calculating Probabilities Using the Normal Approximation | 使用正态近似计算概率
tion is valid and determined the parameters μ = np and σ = √(np(1-p)), follow these steps:
一旦确认正态近似可用，并确定了参数 μ = np 和 σ = √(np(1-p))，即可按以下步骤操作：
- Apply the continuity correction to the binomial probability you need.
- 对所需的二项概率进行连续性修正。
- Standardise the boundary values to obtain z-scores using z = (x - μ) / σ.
- 将边界值标准化，利用 z = (x - μ) / σ 求得 z 分数。
- Use the standard normal distribution table to find the required probability.
- 查阅标准正态分布表以获取所需概率。
For example, if we need P(X ≤ 25) with n=60, p=0.5, we compute μ = 30, σ = √15 ≤ 25.5), standardise to z = (25.5 - 30)/3.873 ≈ -1.16, look up Φ(-1.16) to get the answer.
例如，要求 P(X ≤ 25)，其中 n=60, p=0.5，计算 μ = 30，σ = √15 ≈ 3.873，修正后求 P(Y ≤ 25.5)，标准化得 z = (25.5 - 30)/3.873 ≈ -1.16，查表得 Φ(-1.16) 即可得出答案。
5. Worked Example: Normal Approximation | 实例：正态近似
es. Find the approximate probability of obtaining between 35 and 45 heads inclusive. Let X ~ 5). Check conditions: np = 40, n(1-p) = 40, both > 5, so normal approximation is suitable.
抛掷一枚硬币 80 次。求获得 35 至 45 次正面（含）的近似概率。设 X ~ B(80, 0.5). 检查条件：np = 40, n(1-p) = 40，均大于 5，正态近似适用。
We approximate X by Y ~ continuity correction: P(34.5 ≤ Y ≤ 45.5). Standardise the lower bound: z₁ = (34.5 - 40)/√20 = -5.5/4.472 ≈ -1.230. Upper bound: z₂ = (45.5 - 40)/√20 ≈ 1.230. Then probability = Φ(1.230) - Φ(-1.230) = 2Φ(1.230) - 1 ≈ 2×0.8907 - 1 = 0.7814.
用 Y ~ N(40, 20) 近似 X。需要求 P(35 ≤ X ≤ 45)。连续性修正后为 P(34.5 ≤ Y ≤ 45.5)。标准化下限：z₁ = (34.5 - 40)/√20 = -5.5/4.472 ≈ -1.230。上限：z₂ = (45.5 - 40)/√20 = 5.5/4.472 ≈ 1.230。概率 = Φ(1.230) - Φ(-1.230) = 2Φ(1.230) - 1 ≈ 2×0.8907 - 1 = 0.7814。
The exact binomial probability is approximately 0.782 (to 3 d.p.), showing excellent agreement.
精确二项概率约为 0.782（保留三位小数），吻合度极好。
6. Conditions for the Poisson Approximation | 泊松近似的条件
When n is large and p is small (typically np < 10 and n > 50, though simpler rules like 'n is large, p < 0.1' . This is extremely useful for modelling rare events such as the number of defective items in a large batch, or the number of accidents in a given time period.
当 n 很大而 p 很小（通常 np < 10 且 n > 50，也有简化为 n 大、p < 0.1 的用法），二项分布可用参数 λ = np 的泊松分布来近似。这对模拟稀有事件极为有用，比如大批产品中的缺陷品数量，或特定时间段内的事故数。
n n is large and p is small ensures that λ is moderate and the events resemble a Poisson process.
近似泊松分布为 Po(λ)，其中 λ = np。n 大、p 小的条件确保了 λ 大小适中，且事件发生模式接近泊松过程。
7. Poisson Approximation in Action | 泊松近似实操
For X ~ probabilities can be read from Poisson probability tables or calculated using P(X = k) = e-λ λ^k / k!.
对于满足泊松条件的 X ~ B(n, p)，令 X ≈ Po(λ)，其中 λ = np。然后可通过泊松概率表或公式 P(X = k) = e-λ λ^k / k! 来求取各概率。
The main advantage is that the Poisson formula avoids large binomial coefficients and powers, making hand calculations feasible. No continuity correction is required because we are already mapping to a discrete distribution.
其主要优势在于泊松公式避开了庞大的二项式系数和幂运算，使手算成为可能。由于已映射到离散分布，无需进行连续性修正。
defectives in production (p = 0.02, n = 200), misprints per page (p very small, n large).
常见情境：生产缺陷品（p = 0.02, n = 200），每页印刷错误（p 极小，n 大）。
8. Worked Example: Poisson Approximation | 实例：泊松近似
03, so λ = np = 3. The conditions are met (n large, p small). X ≈ Po(3). Then P(X = 2) = e-3 × 32 / 2! = 0.0498 × 9 / 2 ≈ 0.2240.
某批 1000 只灯泡中，单只次品率为 0.003。求恰有 2 只次品的近似概率。此处 n=1000, p=0.003，所以 λ = np = 3。满足条件（n 大，p 小）。X ≈ Po(3)。则 P(X = 2) = e-3 × 32 / 2! = 0.0498 × 9 / 2 ≈ 0.2240。
The exact binomial probability P(X=2) = C(1000,2) × 0.0032 × 0.997998 , but the Poisson approximation gives a fast and accurate estimate (exact value ≈ 0.2241).
精确二项概率 P(X=2) = C(1000,2)×0.0032×0.997998 手算极难，但泊松近似给出了快速而准确的估计（精确值约 0.2241）。
If the question asked for P(X > 2), we'd use P(X > 2) = 1 - P(X ≤ 2) with Poisson table values.
若题目要求 P(X > 2)，则利用 P(X > 2) = 1 - P(X ≤ 2) 并查泊松表值。
9. Comparing Exact Binomial with Approximations | 精确二项与近似的比较
It's instructive to see how the approximations perform under different parameter settings. The table below compares exact binomial probabilities with normal and Poisson approximations for a few cases.
观察不同参数设置下各近似的表现很有教育意义。下表比较了若干情况下的精确二项概率与正态、泊松近似结果。
| Case (n, p) | Probability | Exact Binomial | Normal Approx. | Poisson Approx. |
|---|---|---|---|---|
| (100, 0.5) | P(45 ≤ X ≤ 55) | 0.7287 | 0.7287 | N/A (λ=50 too large) |
| (200, 0.02) | P(X = 4) | 0.1960 | 0.1959 (with cc) | 0.1954 (λ=4) |
| (500, 0.008) | P(X ≤ 2) | 0.2381 | N/A (p too small) | 0.2381 (λ=4) |
Notice that the normal approximation shines when p is near 0.5, while Poisson excels for very small p. Choosing the wrong approximation can lead to significant errors.
可见，当 p 接近 0.5 时正态近似表现亮眼，而当 p 极小时泊松近似更为出色。选错近似会导致显著误差。
10. Choosing the Right Approximation | 选择合适的近似
In an exam, you must decide whether to use the normal or Poisson approximation, or even to calculate the binomial probability directly if n is small enough. Here is a decision guide:
在考试中，你必须决定是使用正态近似、泊松近似，还是在 n 足够小的情况下直接计算二项概率。决策指南如下：
- lly n ≤ 50) and calculations are manageable, use exact binomial with calculator or tables.
- 若 n 较小（通常 n ≤ 50）且计算量可控，使用计算器或表格求精确二项概率。
- If n is large, and both np > 5 and n(1-p) > 5, use the normal approximation with continuity correction.
- 若 n 大，且 np > 5 与 n(1-p) > 5 均成立，使用带连续修正的正态近似。
- e and p is small (usually p < 0.1 and np < 10), use the Poisson approximation with λ = np.
- 若 n 大且 p 小（通常 p < 0.1 且 np < 10），使用 λ = np 的泊松近似。
- When both sets of conditions partially overlap, Edexcel questions often specify which method to use, or the context makes the choice clear.
- 当两组条件有部分重叠时，爱德思考题通常会指定所用方法，或根据上下文可明确选择。
Always state your checks explicitly in the answer to justify your choice of approximation.
在答案中务必清晰陈述条件检查，以证明所选近似方案的合理性。
11. Common Pitfalls and Tips | 常见错误与技巧
Many students lose marks not because they cannot do the maths, but because they forget a crucial step. Here are the top pitfalls:
许多学生丢分并非因为不懂数学，而是遗漏了关键步骤。常见陷阱如下：
- Forgetting continuity correction when using the normal approximation. Always adjust by 0.5.
- 进行正态近似时忘记连续性修正。务必进行 0.5 的调整。
- Applying the normal approximation when np or n(1-p) is too small. Always check conditions first.
- 在 np 或 n(1-p) 过小时仍使用正态近似。务必先检查条件。
- Confusing the variance: for normal approximation it's np(1-p), not np.
- 混淆方差：正态近似的方差是 np(1-p)，而非 np。
- Using Poisson approximation when np is large (e.g. >10) without checking suitability; Poisson tables may be limited.
- 在 np 较大（如 >10）时未加判断就使用泊松近似；泊松分布表可能有所限制。
- Incorrectly rounding z-scores or using the wrong tail of the normal table.
- 错误舍入 z 分数或使用正态分布表的错误尾部。
Practice with past papers, and always write a concluding statement that interprets the probability in context.
多练习往年试题，并养成写出结论、在情境中解释概率的习惯。
12. Summary and Exam Advice | 总结与考试建议
Approximating a binomial distribution is a powerful technique that tests your understanding of both the binomial model and the properties of normal and Poisson distributions. Remember the key flow: Identify the binomial parameters, check conditions, select the correct approximation, apply corrections if necessary, and compute carefully using standardised values or Poisson formulas.
二项分布的近似是一项强大技术，考察你对二项模型以及正态、泊松分布性质的理解。记住关键流程：识别二项参数，检查条件，选择正确近似，必要时进行修正，并利用标准化值或泊松公式谨慎计算。
In the Edexcel A-Level exam, such questions often carry 6–10 marks and can appear in both the pure statistics and applied contexts. Showing your working for the continuity correction and condition checks is essential for full marks.
在爱德思 A-Level 考试中，此类题目通常占 6 至 10 分，既可能出现在统计部分，也可能在应用题中。展示连续性修正和条件检查的步骤对拿满分至关重要。
With thorough practice, you will be able to tackle any approximation problem confidently and accurately.
经过扎实练习，你一定能自信而准确地解决任何近似问题。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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