Binomial Cumulative Distribution Function | 二项累积分布函数

📚 Binomial Cumulative Distribution Function | 二项累积分布函数

In A‑Level Statistics, the binomial distribution models the number of successes in a fixed number of independent trials. Once you understand how to calculate the probability of exactly x successes using the binomial probability mass function, the next natural step is to ask about cumulative outcomes – for instance, the probability of at most 3 successes, or fewer than 10 defective items. This is where the binomial cumulative distribution function becomes essential. In this article we explore its definition, how to compute it using both tables and calculators, and how to avoid common pitfalls in Edexcel exam questions.

在 A‑Level 统计学中，二项分布用于描述固定次数独立试验中成功的次数。当你掌握了使用二项概率质量函数计算恰好 x 次成功概率的方法后，自然会追问累积的结果——例如至多 3 次成功的概率，或少于 10 件次品的概率。此时二项累积分布函数便显得至关重要。本文将探讨它的定义、如何借助统计表和计算器求得结果，以及如何避开 Edexcel 考试中的常见陷阱。

1. Recalling the Binomial Distribution | 回顾二项分布

Before diving into cumulative probabilities, it is worth refreshing the basic setup. A binomial experiment consists of n independent trials, each with exactly two possible outcomes – success and failure. The probability of success, p, remains constant for every trial. If the random variable X represents the number of successes, we write X ~ B(n, p dividual point probabilities are found using the binomial probability mass function: P(X = x) = nCx · p^x · (1 – p)^{n – x}. This formula is the foundation for building cumulative sums.

深入累积概率之前，有必要回顾一下基本设定。二项试验由 n 次独立试验组成，每次试验只有成功和失败两种可能结果。成功的概率 p 在每次试验中保持不变。若随机变量 X 表示成功的次数，我们记作 X ~ B(n, p)。单点概率通过二项概率质量函数求出：P(X = x) = nCx · p^x · (1 – p)^{n – x}。此公式是构建累积和的基础。

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2. What Does 'Cumulative Probability' Mean? | “累积概率”是什么意思？

In probability theory, a cumulative probability answers questions of the form “what is the chance that the random variable does not exceed a certain value?” For a discrete variable like a binomial count, P(X ≤ x) is the sum of all point probabilities from X = 0 up to X = x. Graphically, you can think of it as the total area of the first (x + 1) bars in a binomial bar chart. The concept extends naturally to statements like P(X < x) and P(X > x), which we will unpack later.

在概率论中，累积概率回答的是“随机变量不超过某个值的概率是多少”这类问题。对于像二项计数这样的离散变量，P(X ≤ x) 是从 X = 0 一直到 X = x 所有单点概率的总和。从图形上看，你可以把它想象成二项条形图中前 (x + 1) 根柱子的总面积。这一概念自然延伸到 P(X < x) 和 P(X > x) 等表达，我们将在后文展开。

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3. Introducing the Binomial Cumulative Distribution Function (CDF) | 二项累积分布函数简介

tion function, often abbreviated as binomial CDF, is a function that directly yields P(X ≤ x) for a given binomial random variable X ~ B(n, p). It is sometimes denoted as F(x) = P(X ≤ x). While the probability mass function gives the height of each bar, the CDF gives the accumulated height up to a chosen value. In Edexcel notation, questions may ask you to find “the probability that X is at most k”, which translates to calculating the binomial CDF at k.

二项累积分布函数常简称为二项 CDF，是一个直接给出 P(X ≤ x) 的函数，其中 X ~ B(n, p)。有时记作 F(x) = P(X ≤ x)。如果说概率质量函数给出每一根柱子的高度，那么 CDF 给出的则是直至选定值为止的累积高度。在 Edexcel 的表述中，题目可能要求你求“X 至多为 k 的概率”，这就转化为在 k 处计算二项 CDF。

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4. Formula and Notation | 公式与记号

The formal definition of the binomial CDF is a simple sum that brings together the binomial coefficients and powers. It is written as:

P(X ≤ x) = Σₖ₌₀ˣ nCₖ · pᵏ · (1 – p)ⁿ⁻ᵏ

Here, the index k runs from 0 to the integer x. The term nCₖ represents the binomial coefficient, which can be computed as n! / [k! (n – k)!] or read directly from Pascal’s triangle or a calculator. In Edexcel examinations you are not required to expand a long sum by hand; instead you will either use printed statistical tables or the cumulative function on your calculator.

二项 CDF 的形式化定义是一个简单的和，将二项式系数与幂次组合在一起。写作：

P(X ≤ x) = Σₖ₌₀ˣ nCₖ · pᵏ · (1 – p)ⁿ⁻ᵏ

这里，索引 k 从 0 取到整数 x。nCₖ 表示二项式系数，可通过 n! / [k! (n – k)!] 计算，也可直接从帕斯卡三角形或计算器读取。在 Edexcel 考试中，你不必手算一长串和；相反，你将使用印刷版统计表或计算器中的累积函数。

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5. Using Binomial Cumulative Distribution Tables | 使用二项累积分布表

Edexcel provides cumulative binomial tables for a range of n and p values in the formula booklet. Each table is typically headed with n and p, and the body lists values of x against the probability P(X ≤ x). To read a table, locate the row corresponding to your x (or less) and read off the decimal. For example, if n = 10, p s 0.8497. Always check that you are using the correct table because separate tables exist for different p values, typically 0.05, 0.10, 0.15, …, 0.50.

Edexcel 在公式手册中提供了不同 n 和 p 组合的二项累积表。每张表通常以 n 和 p 作为标题，表中列出 x 对应的概率 P(X ≤ x)。查表时，找到与你所需 x 对应的行，读取小数概率。例如 n = 10, p = 0.3，需要求 P(X ≤ 4)，你找到 n=10, p=0.30 的表格，移至 x = 4 那一行，便可读取类似 0.8497 的数值。务必确认正在使用的表格正确无误，因为不同的 p 值（如 0.05, 0.10, 0.15, …, 0.50）分别对应不同的表格。

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6. Probabilities for X > x, X ≥ x, and Intervals | 大于、不小于及区间概率

Exam questions rarely ask only for P(X ≤ x). You will often need P(X ≥ a), P(X > a or the probability that X lies between two values. Thanks to the complementary rule, P(X ≥ a) = 1 – P(X ≤ a – 1). Similarly, P(X > a) = 1 – P(X ≤ a). For an interval, P(a ≤ X ≤ b) = P(X ≤ b) – P(X ≤ a – 1). These relationships allow you to extract virtually any needed probability from a single CDF table or calculator output. Always pay attention to the difference between strict and non‑strict inequalities, as missing a single integer can change the answer.

考试题目极少只要求 P(X ≤ x)。你常常需要求 P(X ≥ a)、P(X > a) 或 X 介于两个值之间的概率。借助互补规则，P(X ≥ a) = 1 – P(X ≤ a – 1)。类似地，P(X > a) = 1 – P(X ≤ a)。对于区间，P(a ≤ X ≤ b) = P(X ≤ b) – P(X ≤ a – 1)。这些关系使你能够从一张 CDF 表或计算器输出中提取几乎任何所需概率。始终注意严格不等号与非严格不等号的区别，漏掉一个整数就可能改变答案。

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7. Handling p > 0.5 – The Symmetry Trick | 处理 p > 0.5——对称性技巧

Most printed cumulative binomial tables stop at p = 0.5. So what do you do when the success probability exceeds 0.5? You exploit symmetry: instead of counting successes, count failures. If X ~ B(n, p) with p > 0.5, define q = 1 – p and let Y = n – X be the number of failures. Then Y ~ B(n, q) and q < 0.5. Important conversions: P(X ≤ x) = P(Y ≥ n – x) = 1 – P(Y ≤ n – x – 1). Similarly, P(X ≥ x) = P(Y ≤ n – x). Mastering this trick prevents you from being stuck when a seemingly ‘large’ p appears.

绝大多数印刷版二项累积表只列到 p = 0.5。那么当成功概率超过 0.5 时该怎么办？你可以利用对称性：不数成功的次数，改数失败的次数。若 X ~ B(n, p) 且 p > 0.5，定义 q = 1 – p，并令 Y = n – X 为失败的次数。于是 Y ~ B(n, q)，且 q < 0.5。重要的转换：P(X ≤ x) = P(Y ≥ n – x) = 1 – P(Y ≤ n – x – 1)。类似地，P(X ≥ x) = P(Y ≤ n – x)。掌握这一技巧，当看似“大”的 p 出现时，你便不会束手无策。

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8. Using Technology: Calculator Binomial CDF | 借助技术手段：计算器的二项 CDF

Modern graphing calculators approved for Edexcel exams, such as the Casio fx‑CG50 or TI‑84 Plus, have built‑in binomial CDF functions. You usually navigate to a distribution menu, choose “Binomial CDF” or “Bcd”, and input the values of n, p, and the upper bound x. The output is P(X ≤ x). Most calculators also allow you to input a lower bound, thereby directly calculating P(a ≤ X ≤ b) without manual subtraction. Be careful: calculator syntax varies, so practise with past papers to confirm you are interpreting the menu correctly.

Edexcel 考试允许使用的现代图形计算器（如 Casio fx‑CG50 或 TI‑84 Plus）内置了二项 CDF 函数。通常你需要进入分布菜单，选择“Binomial CDF”或“Bcd”，然后输入 n、p 以及上界 x 的值。输出结果为 P(X ≤ x)。多数计算器还允许输入一个下界，从而直接计算 P(a ≤ X ≤ b)，无需手动相减。需要注意：不同计算器的语法稍有差别，请借助往年真题勤加练习，确保正确理解菜单含义。

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9. Worked Example 1 – At Most k Successes | 例题 1——至多 k 次成功

A biased coin lands heads with probability 0.4. The coin is tossed 12 times. Find the probability that it lands heads at most 5 times.

一枚偏向硬币正面朝上的概率为 0.4。抛掷该硬币 12 次，求正面朝上至多 5 次的概率。

Let X ~ 12, 0.4) be the number of heads. We need P(X ≤ 5). Using the cumulative binomial table for n = 12, p = 0.40, we locate the row with x nction with n=12, p=0.4, x=5 yields approximately 0.6652. This means there is about a 66.5% chance of getting no more than 5 heads in 12 tosses.

令 X ~ B(12, 0.4) 表示正面朝上的次数。我们需要求 P(X ≤ 5)。使用 n = 12, p = 0.40 的二项累积表，找到 x = 5 的行。表格给出 P(X ≤ 5) = 0.6652（数值可能因表格精度稍有出入）。也可以使用计算器 Bcd 功能，输入 n=12, p=0.4, x=5，得到近似值 0.6652。这意味着在 12 次抛掷中，得到不超过 5 次正面的概率约为 66.5%。

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10. Worked Example 2 – Probability of At Least 8 Defects | 例题 2——至少 8 件次品的概率

A factory produces components, and each has a probability of 0.15 of being defective. A random sample of 20 components is taken. Find the probability that at least 8 components are defective.

一家工厂生产零件，每件有 0.15 的概率为次品。随机抽取 20 件样品，求至少 8 件为次品的概率。

Define X ~ he number of defective items. We need P(X ≥ 8). Using the complementary rule, P(X ≥ 8) = 1 – P(X ≤ 7). Since p able for n=20, p=0.15. The table shows P(X ≤ 7) = 0.9941 (approx). Therefore, P(X ≥ 8) = 1 – least 8 defectives out of 20 would be extremely unusual if the defect rate truly is 0.15.

定义 X ~ B(20, 0.15) 为次品数量。我们需要 P(X ≥ 8)。利用互补规则，P(X ≥ 8) = 1 – P(X ≤ 7)。由于 p = 0.15 < 0.5，可直接使用 n=20, p=0.15 的累积表。表格显示 P(X ≤ 7) ≈ 0.9941。因此 P(X ≥ 8) = 1 – 0.9941 = 0.0059。这个概率极小，说明如果次品率真的为 0.15，那么 20 件中出现至少 8 件次品是极不寻常的。

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11. Common Mistakes and How to Avoid Them | 常见错误与规避方法

One typical error is confusing P(X ≤ x) with P(X < x). For discrete variables, P(X < x) = P(X ≤ x – before using tables. Another mistake is forgetting to switch to the failure variable when p > 0.5, leading to table‑lookup errors. Also, when subtracting cumulative probabilities for an interval, double‑check the correct subtraction: P(a ≤ X ≤ b) = P(X ≤ b) – P(X ≤ a – 1). Finally, always check that the sum of probabilities in your reasoning does not exceed 1 – a quick sanity check can catch many slips.

一个典型错误是将 P(X ≤ x) 与 P(X < x) 混淆。对于离散变量，P(X < x) = P(X ≤ x – 1)。在使用表格之前，务必先把严格不等号转换成 ≤ 形式。另一个错误是当 p > 0.5 时忘记切换到失败变量，导致查表失误。此外，在计算区间概率进行相减时，要仔细检查正确的减法：P(a ≤ X ≤ b) = P(X ≤ b) – P(X ≤ a – 1)。最后，始终检查推理中的概率之和是否超过 1——一次快速审核便能发现众多失误。

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12. Summary and Key Takeaways | 总结与关键要点

The binomial cumulative distribution function is an indispensable tool for solving “at most”, “at least” lity problems with binomial data. Key points to remember: (i) the CDF directly gives P(X ≤ x); (ii) all other inequalities can be rewritten using complements or differences; (iii) printed tables typically cover p ≤ 0.5, so use symmetry for larger p; (iv) calculator functions provide a quick and accurate alternative, but must be used with correct syntax. With these skills and plenty of practice, you will handle binomial CDF questions with confidence in your Edexcel examination.

二项累积分布函数是解决二项数据中“至多”“至少”及区间概率问题不可或缺的工具。需记住的要点：(i) CDF 直接给出 P(X ≤ x)；(ii) 其他所有不等号均可通过互补或做差来重写；(iii) 印刷表通常覆盖 p ≤ 0.5，对于更大的 p 要使用对称性；(iv) 计算器函数提供了快捷、准确的替代方案，但务必使用正确的语法。凭借这些技能并多加练习，你将在 Edexcel 考试中自信地应对二项 CDF 相关题目。

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