IB WJEC Mathematics: Probability Key Points | IB WJEC 数学：概率考点精讲

📚 IB WJEC Mathematics: Probability Key Points | IB WJEC 数学：概率考点精讲

Probability is a fundamental pillar of both the IB and WJEC mathematics syllabuses. Whether you are working with tree diagrams, conditional probability, or normal distributions, a solid understanding of the underlying concepts and formulas is essential for exam success. This article breaks down every major probability topic you will encounter, with clear English explanations followed by Chinese translations, so you can master the content in both languages.

概率是 IB 与 WJEC 数学课程的核心支柱之一。无论你面对的是树状图、条件概率还是正态分布，扎实理解基本概念与公式都是考试成功的关键。本文拆解每一个重要的概率考点，提供清晰的英文解释与中文对照，帮助你用双语掌握全部内容。

1. Sample Space and Basic Concepts | 样本空间与基本概念

The sample space is the set of all possible outcomes of an experiment. An event is any subset of the sample space. For a fair six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}, and the event ‘rolling an even number’ is E = {2, 4, 6}. The probability of an event A is given by P(A) = number of favourable outcomes / total number of outcomes, provided all outcomes are equally likely.

样本空间是一次试验中所有可能结果的集合。事件是样本空间的任意子集。对于一枚均匀的六面骰子，样本空间 S = {1, 2, 3, 4, 5, 6}，事件“掷出偶数”为 E = {2, 4, 6}。若所有结果等可能，事件 A 的概率为 P(A) = 有利结果数 / 总结果数。

2. Probability Rules | 概率法则

The probability of any event A satisfies 0 ≤ P(A) ≤ 1. The probability of the complement of A is P(A’) = 1 − P(A). For any two events A and B, the addition rule holds: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B). These rules form the bedrock of almost all probability calculations.

任何事件 A 的概率满足 0 ≤ P(A) ≤ 1。A 的补事件概率为 P(A’) = 1 − P(A)。对于任意两个事件 A 与 B，加法法则成立：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若 A 与 B 互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。这些法则是几乎所有概率计算的基石。

3. Mutually Exclusive and Independent Events | 互斥与独立事件

Two events are mutually exclusive if they cannot occur simultaneously, i.e., A ∩ B = ∅. Two events are independent if the occurrence of one does not affect the probability of the other; mathematically, P(A ∩ B) = P(A)P(B). It is a common mistake to confuse these two concepts: mutually exclusive events are never independent unless one of them has zero probability, because if one occurs, the other cannot.

两个事件若不能同时发生，即 A ∩ B = ∅，则称它们互斥。两个事件若一个的发生不影响另一个的概率，则称它们独立；数学上表达为 P(A ∩ B) = P(A)P(B)。常见的错误是将这两个概念混淆：互斥事件永远不是独立事件（除非其中一个概率为零），因为若一个发生，另一个必定不发生。

4. 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. This formula is used when we have additional information that changes the sample space. Rearranging gives the multiplication rule: P(A ∩ B) = P(B) × P(A | B) or P(A) × P(B | A). Conditional probability is the key to solving problems involving ‘given that’ statements.

事件 A 在事件 B 已发生的条件下的条件概率定义为 P(A | B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。当我们获得额外信息从而改变样本空间时，就要使用该公式。移项可得乘法法则：P(A ∩ B) = P(B) × P(A | B) 或 P(A) × P(B | A)。条件概率是解决所有“已知…时”问题的关键。

5. Tree Diagrams | 树状图

Tree diagrams help visualise sequential events and combine the multiplication and addition rules. Each branch represents a possible outcome, and the probability is written along the branch. For independent events, the probabilities on the second set of branches remain unchanged. For dependent events, the conditional probabilities adjust after the first outcome. To find the probability of a specific final outcome, multiply along the branches; to find the total probability of an event, sum the probabilities of all paths that lead to it.

树状图有助于可视化顺序发生的事件，并结合乘法与加法法则。每条分支代表一个可能结果，概率写在分支上。对于独立事件，第二层分支的概率保持不变。对于相关事件，第一层结果确定后条件概率会相应调整。要求某一最终结果的概率，应沿分支相乘；要求某事件的总概率，应把所有通向该事件的路径概率相加。

6. Venn Diagrams and Set Notation | 文氏图与集合符号

Venn diagrams use overlapping circles within a rectangle to represent sets and their relationships. The rectangle denotes the sample space S. Key notations include A ∩ B (intersection), A ∪ B (union), and A’ (complement). Venn diagrams are particularly useful for visualising and calculating probabilities involving intersection, union, and complements, especially when solving problems with ‘at least one’ or ‘only’ conditions.

文氏图用矩形内的重叠圆形来表示集合及其关系。矩形代表样本空间 S。关键的符号有 A ∩ B（交集）、A ∪ B（并集）以及 A’（补集）。文氏图对可视化及计算涉及交集、并集和补集的概率尤其有用，尤其是在解决“至少一个”或“仅有”条件的问题时。

7. Bayes’ Theorem | 贝叶斯定理

Bayes’ theorem connects conditional probabilities in the reverse direction. It states that P(A | B) = [P(B | A) × P(A)] / P(B). Using the law of total probability, P(B) can be expanded as P(B | A)P(A) + P(B | A’)P(A’) when A and A’ form a partition. This theorem appears regularly in IB and WJEC questions involving diagnostic tests or two‑stage experiments where you need to ‘reverse’ a conditional probability.

贝叶斯定理将条件概率反向关联起来。其形式为 P(A | B) = [P(B | A) × P(A)] / P(B)。利用全概率公式，当 A 与 A’ 构成完备划分时，P(B) 可展开为 P(B | A)P(A) + P(B | A’)P(A’)。该定理经常出现在 IB 和 WJEC 的题目中，涉及诊断测试或需要“逆算”条件概率的两阶段试验。

8. Discrete Random Variables | 离散随机变量

A discrete random variable X takes a countable set of values, each with an associated probability P(X = x). The probability distribution must satisfy Σ P(X = x) = 1. You can represent the distribution using a table showing each value and its probability. The expected value E(X) = Σ x·P(X = x) gives the theoretical mean, while the variance Var(X) = E(X²) − [E(X)]² measures spread.

离散随机变量 X 取可数的数值集合，每个值具有相应的概率 P(X = x)。其概率分布必须满足 Σ P(X = x) = 1。可用表格展示每个取值及其概率。其期望值 E(X) = Σ x·P(X = x) 即为理论均值，而方差 Var(X) = E(X²) − [E(X)]² 则衡量离散程度。

9. Binomial Distribution | 二项分布

A binomial distribution arises when there are a fixed number of independent trials, n, each with two possible outcomes (success/failure) and the same probability of success, p. We write X ~ B(n, p). The probability of exactly r successes is:

P(X = r) = ⁿC_r p^r (1 − p)^{n − r}

where ⁿC_r = n! / [r!(n − r)!]. The mean and variance are E(X) = np and Var(X) = np(1 − p). Recognise binomial conditions quickly: fixed n, independent trials, constant p, two outcomes.

当有固定次数 n 的独立试验，每次试验只有两种结果（成功/失败）且成功的概率 p 保持不变时，随机变量服从二项分布，记作 X ~ B(n, p)。恰好成功 r 次的概率为：

P(X = r) = ⁿC_r p^r (1 − p)^{n − r}

其中 ⁿC_r = n! / [r!(n − r)!]。其均值与方差为 E(X) = np，Var(X) = np(1 − p)。要快速识别二项分布条件：试验次数 n 固定、试验间独立、p 恒定、只有两种结果。

10. Normal Distribution | 正态分布

The normal distribution is a continuous probability distribution with a bell-shaped curve. If X ~ N(μ, σ²), then μ is the mean and σ² the variance. To calculate probabilities, we standardise using Z = (X − μ) / σ, where Z ~ N(0, 1). The standard normal table then gives P(Z < z). For symmetrical intervals, use P(−k < Z < k) = 2P(Z < k) − 1. JWEC and IB exams may include inverse normal calculations to find unknown means or standard deviations.

正态分布是一种具有钟形曲线的连续型概率分布。若 X ~ N(μ, σ²)，则 μ 为均值，σ² 为方差。计算概率时，先进行标准化 Z = (X − μ) / σ，此时 Z ~ N(0, 1)。标准正态分布表可给出 P(Z < z)。对于对称区间，使用 P(−k < Z < k) = 2P(Z < k) − 1。WJEC 与 IB 考试可能会出现逆正态计算，用于求未知的均值或标准差。

11. Expectation and Variance | 期望与方差

For any random variable X and constants a, b, linear transformations work as follows: E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X). For combined random variables, if X and Y are independent, E(X ± Y) = E(X) ± E(Y) and Var(X ± Y) = Var(X) + Var(Y). These properties often help simplify complex probability problems, particularly in WJEC statistics sections.

对于任意随机变量 X 及常数 a、b，线性变换有如下性质：E(aX + b) = aE(X) + b，Var(aX + b) = a²Var(X)。对于多个随机变量，若 X 与 Y 独立，则 E(X ± Y) = E(X) ± E(Y)，Var(X ± Y) = Var(X) + Var(Y)。这些性质在简化复杂的概率问题时非常有用，尤其在 WJEC 统计部分。

12. Exam Tips and Common Mistakes | 考试技巧与常见错误

Always define your random variable clearly at the start. When reading a question, underline key phrases like ‘conditional’, ‘independent’, ‘at least’, and ‘given that’. Check whether probabilities need to be expressed as fractions, decimals or percentages. Avoid confusing P(A | B) with P(A ∩ B); the conditional probability is only meaningful if the conditioning event has occurred. In binomial problems, confirm that trials are independent and p is constant. For normal distribution questions, draw a sketch and label the mean and the required region. Lastly, show all your working step by step — marks are allocated for method as well as the final answer.

解题时务必在一开始就清晰地定义随机变量。阅读题干时，划出“条件”、“独立”、“至少”、“已知”等关键词。注意题目要求概率以分数、小数还是百分数表示。避免将 P(A | B) 与 P(A ∩ B) 混淆；条件概率只有在条件事件已发生时才有意义。处理二项分布问题时，应确认各次试验独立且 p 恒定。在正态分布题中，先画出草图并标出均值与所求区域。最后，解题过程要逐步展示——评分标准既看方法也看最终答案。

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