Mastering Exercise 24D.1: Normal Approximation to Binomial Distribution | 掌握练习24D.1：二项分布的正态近似

📚 Mastering Exercise 24D.1: Normal Approximation to Binomial Distribution | 掌握练习24D.1：二项分布的正态近似

In IB Mathematics, Exercise 24D.1 typically introduces the normal approximation to the binomial distribution — a powerful tool that simplifies probability calculations when the number of trials is large. This technique not only saves time but also deepens your understanding of how discrete distributions can be modelled by continuous ones under certain conditions. In this article, we will explore the key ideas behind the normal approximation, learn when and how to apply the continuity correction, and walk through worked examples to ensure you can tackle any problem from Exercise 24D.1 with confidence.

在IB数学中，练习24D.1通常介绍二项分布的正态近似——当试验次数较大时，这是一个简化概率计算的强大工具。这一技巧不仅能节省时间，还能加深你对离散分布如何在特定条件下被连续分布建模的理解。本文将探讨正态近似背后的关键思想，学习何时以及如何使用连续性校正，并通过详解例题帮助你自信地解决练习24D.1中的任何问题。

1. Reviewing the Binomial Distribution | 二项分布回顾

in a fixed number of independent trials, each with the same probability of success p. If X ~ probabilities can be calculated using the formula or GDC, this becomes impractical when n is large.

二项分布描述的是在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为 p。若 X ~ B(n, p)，则恰好成功 k 次的概率为 P(X = k) = ⁿCₖ pᵏ (1−p)ⁿ⁻ᵏ。其均值为 μ = np，方差为 σ² = np(1−p)。虽然精确概率可以用公式或图形计算器求得，但当 n 很大时这就变得不切实际了。

2. Why Use a Normal Approximation? | 为何使用正态近似？

ormous factorials and many individual terms. The normal distribution, being continuous, offers a convenient approximation because the shape of a binomial distribution becomes symmetric and bell-shaped as n increases, provided p is not too close to 0 or 1. This is a consequence of the Central Limit Theorem and allows us to use standard normal tables to find probabilities quickly.

对于大 n（例如 n = 200）的二项概率计算涉及巨大的阶乘和很多单项。正态分布作为连续分布提供了一个方便的近似，因为当 p 不太接近 0 或 1 时，随着 n 的增大，二项分布的形状会变得对称并呈钟形。这是中心极限定理的结果，使我们能够使用标准正态分布表快速求出概率。

3. Conditions for the Normal Approximation | 正态近似的条件

better accuracy. Meeting these conditions ensures the binomial distribution is roughly symmetric and the tails are not too skewed.

二项分布的正态近似仅在 np 和 n(1−p) 都足够大时才被认为有效。常见的经验法则是 np > 5 且 n(1−p) > 5，不过有些教材使用 np ≥ 10 且 n(1−p) ≥ 10 以获得更好的准确性。满足这些条件能保证二项分布大致对称，尾部不会过于偏斜。

4. Understanding Continuity Correction | 理解连续性校正

ximated by the area under the normal curve up to 10.5, because the integer 10 corresponds to the interval from 9.5 to 10.5 in the continuous scale. This adjustment significantly improves the accuracy of the approximation.

二项分布是离散的（整数结果），而正态分布是连续的。为了弥合这一差异，我们需要应用连续性校正。例如，二项分布中的事件 X ≤ 10 在近似时对应正态曲线下直至 10.5 的面积，因为整数 10 对应于连续尺度上的区间 9.5 到 10.5。这个调整能显著提高近似的准确性。

5. Continuity Correction Rules | 连续性校正的规则

Below is a summary of how to adjust the binomial probability statement for the normal approximation:

以下是调整二项概率语句用于正态近似的总结：

Binomial probability	Normal approximation uses
P(X < k)	P(X < k − 0.5)
P(X ≤ k)	P(X < k + 0.5)
P(X > k)	P(X > k + 0.5)
P(X ≥ k)	P(X > k − 0.5)
P(X = k)	P(k − 0.5 < X < k + 0.5)

Always remember to standardise the corrected boundary to a z-score before using normal tables. The continuity correction reduces the error introduced by replacing a discrete mass with a continuous interval.

在使用正态表之前，务必先将校正后的边界值标准化为 z 分数。连续性校正减小了用连续区间替代离散质量所引入的误差。

6. Standardising to the z-Score | 标准化为 z 分数

(x − μ) / σ. For example, to approximate P(X ≤ 20) for B(50, 0.4), we compute μ = 20, σ ≈ √12 20.5 − 20) / 3.464 ≈ 0.144. Then P(Z < 0.144) can be found from the standard normal table.

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应用连续性校正后，我们使用 μ = np 和 σ = √(np(1−p)) 将边界值转换为标准正态 z 分数。公式为 z = (x − μ) / σ。例如，对于 B(50, 0.4) 近似计算 P(X ≤ 20)，我们计算 μ = 20，σ ≈ √12 ≈ 3.464，使用校正边界 20.5，因此 z = (20.5 − 20) / 3.464 ≈ 0.144。然后可从标准正态表中查出 P(Z < 0.144)。

7. Worked Example: Lower-Tail Probability | 例题：下尾概率

timate P(X ≤ 80) using a normal approximation. Here n=150, p=0.6, so μ = 150×0.6 = 90, σ = √150×0.6×0.4 = √36 0.0567. The exact binomial probability is about 0.0573, so the approximation is excellent.

一枚偏倚硬币出现正面的概率为 0.6，将其抛掷 150 次。设 X 为正面朝上的次数。使用正态近似估计 P(X ≤ 80)。这里 n=150，p=0.6，故 μ=90，σ=6。条件 np=90>5 且 n(1−p)=60>5 成立。应用连续性校正：P(X ≤ 80) 变为 P(X < 80.5)。然后 z = (80.5 − 90)/6 ≈ −1.583。查表得 P(Z < −1.583) ≈ 0.0567。精确的二项概率约为 0.0573，近似效果极好。

8. Worked Example: Probability of Exactly k Successes | 例题：恰好 k 次成功的概率

To approximate P(X = 28) for X ~ B(80, 0.35), we first check conditions: μ = 80×0.35 = 28; σ = √80×0.35×0.65 ≈ √18 32 (using symmetry). The exact binomial value is 0.0936, demonstrating a very close match.

对 X ~ B(80, 0.35) 近似计算 P(X = 28) 时，首先检查条件：μ=28，σ≈4.266，np 和 n(1−p) 均大于 5。利用等式规则，P(X = 28) ≈ P(27.5 < X < 28.5)。计算 z₁ ≈ −0.1172，z₂ ≈ 0.1172。于是 P(−0.1172 < Z < 0.1172) = 2 × 0.0466 = 0.0932（利用对称性）。精确二项值为 0.0936，非常接近。

9. Comparing Approximation with Exact Values | 近似值与精确值的比较

The accuracy of the normal approximation improves with increasing n and when p is near 0.5. For skewed distributions (p close to 0 or 1), the approximation may be poor even if np > 5, but continuity correction still helps. In IB exams, you are often asked to compare the approximate result with the exact value from a GDC or binomial table to discuss the validity of the approximation.

正态近似的准确性随着 n 增大和 p 接近 0.5 而提高。对于偏斜分布（p 接近 0 或 1），即使 np > 5，近似效果也可能不佳，但连续性校正仍有所助益。在 IB 考试中，经常要求你将近似结果与 GDC 或二项分布表中的精确值进行比较，以讨论近似的有效性。

10. Common Pitfalls in Normal Approximation | 正态近似中常见的错误

uality). Always double-check the direction of the inequality and the corresponding ±0.5 rule. Also, ensure the conditions np > 5 and n(1−p) > 5 are verified before approximating.

学生们常常忘记应用连续性校正，这会导致明显的误差，尤其是对于窄区间。另一个常见错误是使用了错误的边界调整，例如将 P(X < k) 写成了 P(X < k − 0.5) 以外的形式。务必反复检查不等式的方向和对应的 ±0.5 规则。此外，在近似之前应确保验证了 np > 5 和 n(1−p) > 5 的条件。

11. Tactic for Exercise 24D.1 Style Questions | 应对练习24D.1型题目的策略

Exercise 24D.1 problems typically provide values of n and p and ask you to calculate binomial probabilities approximately, then compare with exact values. Begin by writing down μ and σ. State that np and n(1−p) satisfy the condition. Rewrite the probability statement with continuity correction. Standardise and obtain the probability from the standard normal table or calculator. Finally, compute the exact binomial probability using technology and comment on the difference. Showing all steps clearly will earn full marks in IB assessment.

练习24D.1 的题目通常会给出 n 和 p 的值，要求你近似计算二项概率，然后与精确值进行比较。首先写下 μ 和 σ。说明 np 和 n(1−p) 满足条件。用连续性校正改写概率语句。进行标准化并从标准正态表或计算器中得到概率。最后使用技术手段计算精确的二项概率，并评论两者的差异。清晰展示所有步骤将在 IB 评估中获得满分。

12. Summary and Final Tips | 总结与最后建议

Mastering the normal approximation to the binomial distribution is essential for IB Mathematics and Exercise 24D.1. The key steps are: check np and n(1−p), apply the correct continuity correction, standardise, and interpret the result. This technique not only simplifies complex calculations but also illustrates a fundamental bridge between discrete and continuous probability. Practice with varied p values and n sizes to build confidence, and always verify approximations against technology when possible.

掌握二项分布的正态近似对 IB 数学和练习24D.1 至关重要。关键步骤是：检查 np 和 n(1−p)，应用正确的连续性校正，标准化并解读结果。这一技巧不仅简化了复杂计算，还展示了离散概率与连续概率之间的基本联系。通过练习不同的 p 值和 n 值来建立自信，并在可能时始终用技术手段验证近似结果。

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