A-Level OCR Maths: Probability Revision Guide | 概率考点精讲

📚 A-Level OCR Maths: Probability Revision Guide | 概率考点精讲

Probability is a core topic in the OCR A-Level Mathematics specification, underpinning statistical inference and decision-making. Mastering the fundamental concepts, rules, and distributions allows students to model real-world uncertainty and solve exam problems confidently.

概率是OCR A-Level数学大纲的核心主题，是统计推断和决策的基础。掌握基本概念、规则和分布，能让学生对现实世界的不确定性进行建模，并自信地解决考试题目。

1. Basic Probability Concepts | 基本概率概念

The probability of an event A, denoted P(A), is a number between 0 and 1 that measures the likelihood of the event occurring. The sample space S contains all possible outcomes, and P(S) = 1.

事件A的概率，记作P(A)，是一个介于0到1之间的数，衡量事件发生的可能性。样本空间S包含所有可能的结果，且P(S)=1。

The complement of A, written A’ or A^c, satisfies:

P(A’) = 1 – P(A)

补集A’满足：

P(A’) = 1 – P(A)

For any event, 0 ≤ P(A) ≤ 1. The impossible event ∅ has probability 0.

任何事件都满足0 ≤ P(A) ≤ 1。不可能事件∅的概率为0。

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2. Mutually Exclusive and Independent Events | 互斥事件与独立事件

Two events A and B are mutually exclusive if they cannot occur at the same time: P(A ∩ B) = 0. The addition rule for mutually exclusive events is:

P(A ∪ B) = P(A) + P(B)

如果两个事件A和B不能同时发生，则它们互斥：P(A ∩ B) = 0。互斥事件的加法法则是：

P(A ∪ B) = P(A) + P(B)

If events are not mutually exclusive, we use the general addition rule:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

如果事件不互斥，则使用一般加法法则：

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Events A and B are independent if the occurrence of one does not affect the probability of the other. For independent events:

P(A ∩ B) = P(A) × P(B)

如果事件A的发生不影响事件B的概率，则A和B独立。对于独立事件：

P(A ∩ B) = P(A) × P(B)

Note: mutually exclusive events with non-zero probabilities are never independent, because if one happens the other cannot.

注意：具有非零概率的互斥事件绝不独立，因为如果一个发生，另一个就不可能发生。

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3. Conditional Probability | 条件概率

The conditional probability of A given B is defined as:

P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0

给定B时A的条件概率定义为：

P(A|B) = P(A ∩ B) / P(B)，前提P(B) > 0

Rearranging gives the multiplication rule: P(A ∩ B) = P(A|B) P(B) = P(B|A) P(A). This is essential for tree diagrams.

移项可得乘法法则：P(A ∩ B) = P(A|B) P(B) = P(B|A) P(A)。这对树形图至关重要。

For independent events, P(A|B) = P(A) and P(B|A) = P(B). Bayes’ theorem is often applied in more complex problems:

P(A|B) = [P(B|A) P(A)] / P(B)

对于独立事件，P(A|B)=P(A)且P(B|A)=P(B)。贝叶斯定理常用于更复杂的问题：

P(A|B) = [P(B|A) P(A)] / P(B)

OCR exams often require applying conditional probability in context, such as disease testing or weather forecasting.

OCR考试常要求将条件概率应用于实际情境，如疾病检测或天气预报。

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4. Tree Diagrams | 树形图

Tree diagrams are a powerful tool for displaying sequences of events and applying the multiplication and addition rules. Each branch is labelled with a probability; probabilities of successive branches multiply along the path.

树形图是显示事件序列并应用乘法和加法法则的强大工具。每个分支标有概率；沿路径连续分支的概率相乘。

To find the probability of a specific combination, multiply the branch probabilities. To find the total probability of an event that can occur via multiple paths, add the path probabilities.

要找到特定组合的概率，将分支概率相乘；要找到可通过多个路径发生的事件总概率，将路径概率相加。

Always check that the probabilities on branches from the same node sum to 1. Conditional probabilities are easily read from the second level of branches.

务必检查同一节点各分支的概率之和为1。条件概率很容易从第二层分支中读出。

Example: drawing two balls without replacement. The probability of a red then a blue is P(R first) × P(B second | R first).

举例：无放回地抽取两个球。先红后蓝的概率为 P(先红) × P(后蓝|先红)。

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5. Venn Diagrams and Set Notation | 维恩图与集合符号

Venn diagrams visually represent sets and their relationships. You need to use set notation fluently: A ∩ B (intersection), A ∪ B (union), A’ (complement), and ∅ (empty set).

维恩图直观地表示集合及其关系。你需要熟练使用集合符号：A ∩ B（交集），A ∪ B（并集），A’（补集），以及∅（空集）。

Probabilities can be assigned to regions of a Venn diagram. The area representing A ∪ B includes all outcomes in A or B or both.

概率可以分配到维恩图的各个区域。表示A ∪ B的区域包括所有属于A或B或两者的结果。

A common exam task is to complete a Venn diagram from given probabilities and then use it to find conditional probabilities like P(A | B).

常见的考试任务是依据给定概率完成维恩图，然后用它求条件概率，如

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