GCSE CCEA Maths: Probability Revision Guide | GCSE CCEA 数学：概率考点精讲

📚 GCSE CCEA Maths: Probability Revision Guide | GCSE CCEA 数学：概率考点精讲

Probability is one of the most accessible yet deceptive topics in GCSE CCEA Mathematics – students often find the basics straightforward but lose marks on multi‑step problems, conditional probability, or tree diagrams without replacement. This revision guide walks you through every concept tested in the CCEA specification, from foundation probability scales to higher‑tier conditional probability and Venn diagrams.

概率是 GCSE CCEA 数学中既平易近人又容易失分的主题——基础知识不难，但多步问题、条件概率和无放回树形图常常丢分。本文梳理了 CCEA 考纲中所有概率考点，从基础的标尺概念到高阶的条件概率与维恩图，助你系统复习。

1. Basic Probability Concepts | 基本概率概念

Probability measures how likely an event is to occur, always lying between 0 (impossible) and 1 (certain). You can express it as a fraction, decimal, or percentage – for example, a fair coin landing heads has probability 1/2, 0.5, or 50%.

概率衡量事件发生的可能性，取值范围在 0（不可能）到 1（必然）之间。可以用分数、小数或百分数表示，例如抛一枚均匀硬币正面朝上的概率为 1/2、0.5 或 50%。

The notation P(A) represents the probability of event A. The complement of A, written A’, covers all outcomes not in A, and we have P(A’) = 1 − P(A). This is incredibly useful when it is easier to calculate the chance something does not happen.

用 P(A) 表示事件 A 的概率。A 的互补事件记为 A’，包含所有不属于 A 的结果，且满足 P(A’) = 1 − P(A)。当计算“不发生”的概率更容易时，这一性质极为有用。

For equally likely outcomes, the basic formula applies:
P(A) = number of favourable outcomes / total number of possible outcomes.

对于等可能结果，基本公式为：
P(A) = 有利结果数 / 等可能结果总数。

2. Sample Space and Equally Likely Outcomes | 样本空间与等可能结果

A sample space is the set of all possible outcomes of an experiment. For a single fair dice, the sample space is {1, 2, 3, 4, 5, 6}. To find probabilities for combined events like rolling two dice, a sample space diagram (a two‑way table) helps list all 36 equally likely ordered pairs.

样本空间是某试验所有可能结果的集合。掷一枚均匀骰子的样本空间为 {1, 2, 3, 4, 5, 6}。对于掷两枚骰子等组合事件，可用样本空间表（双向表）列出全部 36 个等可能的有序数对。

Dice 1 \ Dice 2	1	2	3	4	5	6
1	(1,1)	(1,2)	(1,3)	(1,4)	(1,5)	(1,6)
2	(2,1)	(2,2)	(2,3)	(2,4)	(2,5)	(2,6)
3	(3,1)	(3,2)	(3,3)	(3,4)	(3,5)	(3,6)
4	(4,1)	(4,2)	(4,3)	(4,4)	(4,5)	(4,6)
5	(5,1)	(5,2)	(5,3)	(5,4)	(5,5)	(5,6)
6	(6,1)	(6,2)	(6,3)	(6,4)	(6,5)	(6,6)

From the table, you can see, for instance, that the probability of scoring a sum of 7 is 6/36 = 1/6 because the favourable outcomes are (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1).

从上表可以看出，点数之和为 7 的概率是 6/36 = 1/6，因为有利结果有 (1,6), (2,5), (3,4), (4,3), (5,2) 和 (6,1)。

Always check whether outcomes are truly equally likely. A spinner with segments of unequal area will not have equally likely outcomes, so a simple count of sections is wrong – you must work with angles or areas.

务必检查结果是否真正等可能。扇形面积不均匀的转盘并不等可能，此时简单数格数就会出错——必须依据角度或面积来计算。

3. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法法则

Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a dice, getting an odd number and getting a 2 are mutually exclusive (you cannot roll both). The addition rule for mutually exclusive events is:
P(A or B) = P(A) + P(B).

若两个事件不能同时发生，则称它们互斥。例如掷骰子时，“得到奇数”和“得到 2”就互斥（不可能同时掷出）。互斥事件的加法法则为：
P(A 或 B) = P(A) + P(B)。

If events are not mutually exclusive, you must subtract the overlap to avoid double counting:
P(A or B) = P(A) + P(B) − P(A and B). This general addition rule is essential for higher‑tier problems, especially those involving Venn diagrams or two‑way tables.

若事件不互斥，则必须减去重叠部分以避免重复计算：
P(A 或 B) = P(A) + P(B) − P(A 且 B)。这一般加法公式对高阶题目至关重要，尤其是涉及维恩图或双向表的题目。

A common CCEA question gives probabilities of a student studying Maths (M) and Physics (P) with some overlap; you would compute P(M ∪ P) = P(M) + P(P) − P(M ∩ P).

CCEA 常见题型会给出学生学习数学 (M) 和物理 (P) 的概率且存在交集，此时需要计算 P(M ∪ P) = P(M) + P(P) − P(M ∩ P)。

4. Independent Events and the Multiplication Rule | 独立事件与乘法法则

Events are independent if the occurrence of one does not affect the probability of the other. Flipping a coin and rolling a dice are independent – the coin’s result does not change the dice probability. For independent events A and B, the multiplication rule applies:
P(A and B) = P(A) × P(B).

如果一事件的发生不影响另一事件的概率，则两事件独立。抛硬币与掷骰子相互独立——硬币结果不改变骰子的概率。对于独立事件 A 和 B，可用乘法法则：
P(A 且 B) = P(A) × P(B)。

Beware: independence is often confused with mutual exclusivity. Mutually exclusive events are never independent (unless one has zero probability) because if one happens, the other cannot happen – so the probability changes.

注意：独立常与互斥混淆。实际上，互斥事件绝不独立（除非某个事件的概率为 0），因为一旦一个事件发生，另一个就不能发生——概率已经改变。

In tree diagrams, events on different branches are often independent (if there is replacement), and you multiply along branches to find combined outcomes. The order of multiplication does not matter because of commutativity.

在树形图中，不同分支上的事件常为独立（有放回时），计算组合结果时沿分支相乘。乘法顺序不影响结果，因为乘法交换律成立。

5. Probability Tree Diagrams | 概率树形图

Tree diagrams are essential for mapping out sequences of events. In CCEA exams, you must be able to draw and complete tree diagrams for both independent and dependent events. Label each branch with its probability – the probabilities from a single point must sum to 1.

树形图是理清事件序列的关键工具。CCEA 考试中，你需要能够绘制并补充独立事件和相依事件的树形图。每条分支标出其概率，从同一点发出的所有分支概率之和必须为 1。

To find the probability of a path, multiply along the branches. For instance, the probability of getting two heads when flipping a fair coin twice is:
P(H and H) = 1/2 × 1/2 = 1/4.

求某条路径的概率，沿分支相乘。例如，抛两次均匀硬币得到两个正面的概率为：
P(正且正) = 1/2 × 1/2 = 1/4。

For without replacement problems, the probabilities on the second set of branches change because the outcomes are no longer independent. If a bag contains 5 red and 3 green sweets and you take two without replacement, the tree must show conditional probabilities such as P(second red | first red) = 4/7.

对于 无放回 问题，第二级分支上的概率会改变，因为结果不再独立。若袋中有 5 颗红色糖和 3 颗绿色糖，无放回抽取两颗，树形图必须显示条件概率，例如 P(第二颗红 | 第一颗红) = 4/7。

When more than one path gives the desired outcome, calculate each path’s probability separately and add them – this uses the intersection‑then‑union approach.

当有多条路径导向同一结果时，分别计算每条路径的概率再相加——这使用了先交后并的思路。

6. Conditional Probability | 条件概率

Conditional probability measures the likelihood of an event occurring given that another event has already happened. The formal notation is P(A|B), read as “probability of A given B”. The key formula is:
P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.

条件概率是在另一事件已发生的前提下，某事件发生的可能性。正式的记法为 P(A|B)，读作“在 B 发生的条件下 A 的概率”。核心公式为：
P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。

This formula appears frequently in higher‑tier CCEA papers. You might be given a Venn diagram or two‑way table and asked to find P(A|B). Simply locate the intersection count (or probability) and divide by the total for event B.

该公式频繁出现在 CCEA 高阶试卷中。你可能会遇到给出维恩图或双向表、要求计算 P(A|B) 的题目。只需找到交集的频数（或概率），再除以事件 B 的总计即可。

From a tree diagram, P(A|B) can be found by taking the probability of the path involving both A and B and dividing by the total probability of all paths that include B. This is essentially Bayes’ mindset at GCSE level.

从树形图求 P(A|B)，可取包含 A 和 B 的路径概率，再除以所有包含 B 的路径总概率。这其实已经是 GCSE 层面的贝叶斯思想。

Example: In a class, 12 students study Art (A) and 20 study Biology (B). 8 study both. Then P(A|B) = 8/20 = 2/5.

举例：某班级有 12 人选修艺术 (A)，20 人选修生物 (B)，8 人两门都选。则 P(A|B) = 8/20 = 2/5。

7. Venn Diagrams and Probability | 维恩图与概率

Venn diagrams illustrate sets and their relationships using overlapping circles inside a rectangle that represents the universal set. They are ideal for solving problems involving “and” (intersection), “or” (union), and “not” (complement), especially when data are given as numbers or probabilities.

维恩图用矩形（全集）内重叠的圆圈表示集合及其关系，非常适合解决涉及“且”（交集）、“或”（并集）和“非”（补集）的概率问题，尤其当数据以频数或概率给出时。

Start by placing the intersection value P(A ∩ B) in the overlapping region, then work outward to fill the remaining parts of A and B, ensuring each region sums correctly. The rectangle outside the circles represents P(A’ ∩ B’).

先将交集值 P(A ∩ B) 填入重叠区域，再向外推算并填充 A 与 B 的剩余部分，确保各区域总和正确。圆圈外、矩形内的部分代表 P(A’ ∩ B’)。

Conditional probabilities are easily read from a Venn diagram: P(A|B) = (number in A ∩ B) / (total in B). Also check that all probabilities in the diagram add up to 1.

从维恩图上可轻松读取条件概率：P(A|B) = (A ∩ B 的频数) / (B 的总频数)。同时要检查图中所有概率之和是否为 1。

A typical CCEA question presents a diagram with numbers inside and asks for probabilities in fraction form – always count the total number of items to get the denominator right.

典型的 CCEA 题目会给出标有数字的维恩图，要求用分数写出概率——务必数清项目总数，确保分母正确。

8. Two‑Way Tables and Frequency Trees | 双向表与频率树

Two‑way tables organise data according to two categories, making them perfect for calculating marginal, joint, and conditional probabilities. Each cell shows a frequency, and marginal totals are found by summing rows or columns.

双向表按两个类别组织数据，非常便于计算边缘概率、联合概率和条件概率。每个单元格为频数，边缘总计可由行或列求和得到。

Consider a table showing 50 students classified by gender and whether they walk to school. The structure immediately reveals, for example, the probability that a randomly chosen student is a boy who walks, or the conditional probability that a student walks given they are a girl.

设想一个表格将 50 名学生按性别和是否步行上学分类。该结构立刻能求出例如随机选一名学生是步行上学男生的概率，或给定是一名女生的条件下该生步行上学的条件概率。

Frequency trees work in a similar way but split outcomes sequentially. Starting with a total number, you branch according to one attribute, then sub‑branch by the second attribute. The final frequencies on the right‑most tips give counts for all combinations, which can be converted to probabilities.

频率树与之类似，但按顺序拆分结果。从总数开始，先按第一属性分支，再按第二属性子分支。最右侧末端的频数给出所有组合的计数，并可转化为概率。

Work methodically: fill in all given frequencies, compute missing ones using mental arithmetic, and only then identify the probability you need.

解题时应条理清晰：填入所有已知频数，利用心算补全缺失值，最后再定位所需概率。

9. Relative Frequency and Expectation | 相对频率与期望值

Relative frequency is an estimate of probability based on experimental data:
Relative frequency = number of successful trials / total number of trials.

相对频率是基于试验数据的概率估计值：
相对频率 = 成功试验次数 / 总试验次数。

As the number of trials increases, the relative frequency tends to get closer to the theoretical probability (the law of large numbers). CCEA questions often ask you to compare an experimental probability from a table of frequencies with the theoretical value and comment on the difference.

随着试验次数增加，相对频率会趋近于理论概率（大数定律）。CCEA 常要求对比频率表给出的实验概率与理论值，并评论其差异。

Expected frequency is the number of times you would expect an event to occur in a given number of trials, calculated by:
Expected frequency = probability × number of trials.

期望频数是在给定试验次数下，预期某事件发生的次数，计算公式为：
期望频数 = 概率 × 试验次数。

For example, if a biased dice has a probability of

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GCSE CCEA Maths: Probability Revision Guide | GCSE CCEA 数学：概率考点精讲