Understanding Probability for IGCSE Mathematics | IGCSE 数学概率核心解析

📚 Understanding Probability for IGCSE Mathematics | IGCSE 数学概率核心解析

Probability is a fundamental branch of mathematics that measures how likely an event is to occur. In the IGCSE syllabus, it appears in Paper 1 and Paper 2, often worth 5–10 marks in total, and it connects naturally to ratio, fractions, and data handling. This article walks through every core concept you need, from basic definitions to conditional probability and tree diagrams, with worked examples that mirror real exam questions.

概率是数学中衡量事件发生可能性大小的基础分支。在 IGCSE 考纲中，概率出现在 Paper 1 和 Paper 2 中，通常占总分 5–10 分，并与比例、分数和数据处理自然衔接。本文将系统梳理从基本定义到条件概率和树状图的全部核心概念，并提供与真实考题风格一致的例题精讲。

1. The Language of Chance | 概率的基本语言

In probability, an experiment is a process that produces an outcome, such as rolling a die or drawing a card. The set of all possible outcomes is called the sample space, usually denoted as S. An event is a subset of the sample space that we are interested in, often labelled with a capital letter like A or B.

在概率中，试验（experiment） 是一个产生结果的过程，例如掷骰子或抽牌。样本空间（sample space） 是所有可能结果的集合，通常记作 S。事件（event）strong> 是样本空间的某个子集，即我们感兴趣的结果，常用大写字母如 A 或 B 表示。

For example, when rolling a fair six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}. If event A is “rolling an even number”, then A = {2, 4, 6}.

例如，掷一枚公平的六面骰子时，样本空间是 S = {1, 2, 3, 4, 5, 6}。若事件 A 表示“掷出偶数”，则 A = {2, 4, 6}。

Sample space: all possible results | 样本空间：所有可能的结果
Event: a specific collection of outcomes | 事件：特定结果的集合
Outcome: a single result of an experiment | 结果：一次试验得到的单一结果

2. The Probability Scale | 概率的数值范围

Probability is always a number between 0 and 1 inclusive. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain. Fractions, decimals, and percentages are all acceptable forms in IGCSE answers.

概率的数值始终在 0 到 1 之间（包含 0 和 1）。概率为 0 表示事件不可能发生，概率为 1 表示事件必然发生。在 IGCSE 考试中，分数、小数和百分数都是可接受的答案形式。

0 ≤ P(A) ≤ 1

For example, the probability of rolling a 7 on a standard die is 0, and the probability of rolling a number less than 7 is 1.

例如，掷一枚标准骰子得到 7 的概率是 0，而得到小于 7 的数字的概率是 1。

Describing probabilities in words is also tested: “impossible” corresponds to 0, “unlikely” to a value from 0 up to 0.5, “even chance” to exactly 0.5, “likely” to from 0.5 up to 1, and “certain” to 1. You should be able to place an event on a probability scale line.

用文字描述概率也是考点：“不可能”对应 0，“不太可能”对应 0 到 0.5 之间的值，“可能性相等”对应恰好 0.5，“很可能”对应 0.5 到 1 之间的值，“必然”对应 1。你需要能够将事件标在概率数轴上。

3. Calculating Basic Probability | 基础概率计算

For a fair experiment where every outcome is equally likely, the probability of an event A is given by:

在每次试验结果等可能的情况下，事件 A 的概率为：

P(A) = n(A) / n(S)

where n(A) is the number of outcomes in event A, and n(S) is the total number of outcomes in the sample space.

其中 n(A) 是事件 A 包含的结果数，n(S) 是样本空间的总结果数。

Worked example: A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. A marble is picked at random. Find the probability that it is blue.

例题：一个袋子中有 3 颗红球、5 颗蓝球和 2 颗绿球。随机取出一颗球，求取出蓝球的概率。

Total marbles = 3 + 5 + 2 = 10. Favorable outcomes = 5. Therefore P(blue) = 5/10 = 1/2 = 0.5.

球的总数 = 3 + 5 + 2 = 10。有利结果数 = 5。因此 P(蓝球) = 5/10 = 1/2 = 0.5。

4. Complementary Events | 互补事件

The complement of event A, written as A’ or Aᶜ, contains all outcomes that are not in A. Since either A or its complement must happen, their probabilities add up to 1:

事件 A 的补事件，记作 A’ 或 Aᶜ，包含所有不属于 A 的结果。由于 A 与其补事件必有一个发生，它们的概率之和等于 1：

P(A) + P(A’) = 1

P(A’) = 1 – P(A)

This is especially useful when the probability of the complement is easier to calculate than the probability of the event itself.

当补事件的概率比事件本身更容易计算时，这个公式非常实用。

Worked example: The probability that it rains on a given day is 0.35. Find the probability that it does not rain.

例题：某天下雨的概率是 0.35。求不下雨的概率。

P(no rain) = 1 – 0.35 = 0.65.

P(不下雨) = 1 – 0.35 = 0.65。

5. 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, rolling a 2 and rolling a 5 on a single die are mutually exclusive because only one side can land face up.

如果两个事件不能同时发生，则称它们为互斥事件（mutually exclusive events）。例如，掷一次骰子得到 2 和得到 5 是互斥的，因为只有一面朝上。

For mutually exclusive events A and B, the probability that A or B occurs is:

对于互斥事件 A 和 B，A 或 B 发生的概率为：

P(A ∪ B) = P(A) + P(B)

Worked example: A card is drawn from a standard deck of 52 cards. Find the probability that it is either a heart or a spade.

例题：从一副 52 张的标准扑克牌中抽一张，求抽到红桃或黑桃的概率。

These events are mutually exclusive (a card cannot be both a heart and a spade). P(heart) = 13/52 and P(spade) = 13/52. Therefore P(heart or spade) = 13/52 + 13/52 = 26/52 = 1/2.

这两个事件互斥（一张牌不可能同时是红桃和黑桃）。P(红桃) = 13/52，P(黑桃) = 13/52。因此 P(红桃或黑桃) = 13/52 + 13/52 = 26/52 = 1/2。

If events are not mutually exclusive, the general addition rule must be used:

如果事件并不互斥，则需使用更一般的加法法则：

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

The subtraction of P(A ∩ B) prevents double-counting outcomes that belong to both events.

减去 P(A ∩ B) 是为了避免重复计算同时属于两个事件的结果。

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

Two events are independent if the occurrence of one does not affect the probability of the other. For independent events A and B, the probability that both occur together is:

如果其中一个事件的发生不影响另一个事件的概率，则称这两个事件为独立事件（independent events）。对于独立事件 A 和 B，两者同时发生的概率为：

P(A ∩ B) = P(A) × P(B)

Worked example: A fair coin is tossed and a fair die is rolled. Find the probability of getting a head on the coin and a 6 on the die.

例题：同时抛一枚公平硬币和掷一枚公平骰子。求硬币正面朝上且骰子掷出 6 的概率。

These events are independent. P(head) = 1/2 and P(6) = 1/6. Therefore P(head and 6) = 1/2 × 1/6 = 1/12.

这两个事件相互独立。P(正面) = 1/2，P(掷出 6) = 1/6。因此 P(正面且 6) = 1/2 × 1/6 = 1/12。

In IGCSE questions, never assume independence unless you are told the events are independent, or the objects are replaced after selection.

在 IGCSE 题目中，除非题目明确说明事件独立，或抽取后放回，否则不可轻易假设独立性。

7. Without Replacement and Tree Diagrams | 不放回抽样与树状图

When objects are selected without replacement, each subsequent probability changes because the sample space shrinks. Tree diagrams are the best tool to handle these multi-stage experiments.

当进行不放回（without replacement）抽取时，每次后续的概率都会因样本空间减小而改变。树状图是处理这类多阶段试验的最佳工具。

Worked example: A box contains 4 red and 6 blue pens. Two pens are drawn at random without replacement. Find the probability that both pens are red.

例题：一个盒子中有 4 支红笔和 6 支蓝笔。随机不放回地抽取 2 支笔。求两支都是红笔的概率。

First draw: P(red) = 4/10. If the first pen is red, there are now 3 red pens left and 9 pens in total. Second draw: P(red | first was red) = 3/9. Therefore P(both red) = 4/10 × 3/9 = 12/90 = 2/15.

第一次抽取：P(红) = 4/10。若第一支是红笔，则剩下 3 支红笔，共 9 支笔。第二次抽取：P(红 | 第一次为红) = 3/9。因此 P(两支都是红笔) = 4/10 × 3/9 = 12/90 = 2/15。

When drawing a tree diagram, label every branch with its probability. The probability of a sequence of events is found by multiplying along the branches, and the probability of a combined event such as “exactly one red pen” is found by adding the probabilities of each path that produces that outcome.

绘制树状图时，须在每条分支上标注概率。某一事件序列的概率通过沿分支相乘得到；而“恰好一支红笔”这类复合事件的概率，则通过将所有符合该结果的路径的概率相加得到。

8. Conditional Probability | 条件概率

Conditional probability asks: given that event B has already occurred, what is the probability of event A? It is written as P(A | B), read as “the probability of A given B”.

条件概率问题关注的是：在事件 B 已经发生的条件下，事件 A 发生的概率是多少？记作 P(A | B)，读作“在 B 条件下 A 的概率”。

P(A | B) = P(A ∩ B) / P(B)

where P(B) is greater than 0.

其中要求 P(B) 大于 0。

Worked example: In a group of students, 15 study French and 10 study Spanish. 4 students study both. A student is chosen at random from those who study Spanish. Find the probability that this student also studies French.

例题：一个学生群体中，15 人学习法语，10 人学习西班牙语，其中 4 人两门都学。从学习西班牙语的学生中随机选一人，求该学生也学习法语的概率。

P(French | Spanish) = (number studying both) / (number studying Spanish) = 4/10 = 2/5.

P(法语 | 西班牙语) = (两门都学的人数) / (学习西班牙语的人数) = 4/10 = 2/5。

In tree diagrams, conditional probabilities are naturally shown on the branches of the second and later stages. Always check carefully whether the phrase “given that” or “if” appears in the question, as this signals a conditional probability.

在树状图中，条件概率自然体现在第二层及之后的分支上。务必留意题目中是否出现 “given that” 或 “if” 等提示条件概率的关键词。

9. Expected Value | 期望值

The expected value (or expectation) of an experiment is the average outcome if the experiment were repeated many times. It is calculated by multiplying each outcome by its probability and adding the results.

期望值是试验在大量重复下结果的平均水平。计算方法是将每个结果乘以它的概率，再将各乘积相加。

E = Σ [x × P(x)]

Worked example: A charity game charges £2 to play. A fair coin is tossed. If it lands heads, the player wins £5; if tails, the player wins nothing. Find the expected winnings per game.

例题：一个慈善游戏收费 2 英镑玩一次。抛一枚公平硬币，若正面朝上玩家赢得 5 英镑；若反面朝上则没有奖励。求每次游戏的期望收益。

Expected winnings = (5 × 1/2) + (0 × 1/2) = £2.50. Since the game costs £2 to play, the expected net gain is 2.50 – 2.00 = £0.50 per game.

期望收益 = (5 × 1/2) + (0 × 1/2) = 2.50 英镑。由于每次游戏收费 2 英镑，所以每场游戏的期望净收益是 2.50 – 2.00 = 0.50 英镑。

Expectation questions often appear in the context of games or experiments in Paper 2. Notice that the expected value need not be one of the actual outcomes; it is a long-run average.

期望值类问题常以游戏或实验为背景出现在 Paper 2 中。注意期望值不必是实际结果中的某一个，它只是一个长期平均值。

10. Common Exam Pitfalls and Tips | 常见易错点与考场建议

The most common mistakes in IGCSE probability questions include confusing mutually exclusive with independent events, forgetting to update probabilities in without-replacement scenarios, and adding probabilities instead of multiplying them for independent events.

IGCSE 概率题中最常见的错误包括：混淆互斥事件与独立事件、在无放回情境中忘记更新概率、以及在独立事件中误用加法而非乘法。

Before answering, always ask yourself three questions. First, are the events equally likely? Second, is the selection with or without replacement? Third, does the question ask for “and” (multiply) or “or” (add)?

作答前，务必自问三个问题：第一，各个事件是否等可能？第二，抽样是放回还是不放回？第三，题目要求的是“且”（相乘）还是“或”（相加）？

Finally, give your answer as a fraction in its simplest form in IGCSE unless the question requests a decimal or percentage. Show your full working for method marks even if your final answer is incorrect.

最后，在 IGCSE 中除非题目要求小数或百分数，答案应以最简分数形式书写。即使最终结果有误，也要展示完整的解题过程以便获得方法分。

11. Quick Reference Summary | 速查总结

Rule	Formula	Use it when…
Complement	P(A’) = 1 – P(A)	You need “not A”
Addition (mutually exclusive)	P(A ∪ B) = P(A) + P(B)	Events cannot overlap
Addition (general)	P(A ∪ B) = P(A) + P(B) – P(A ∩ B)	Events can overlap
Multiplication (independent)	P(A ∩ B) = P(A) × P(B)	One event does not affect the other
Conditional probability	P(A \| B) = P(A ∩ B) / P(B)	You know B has already happened
Expected value	E = Σ [x × P(x)]	Finding long-run average

Keep this table in your revision notes, because it gathers every core probability formula in one place. Master each rule with a fresh example, and you will be well prepared for IGCSE questions on this topic.

将这张表格收入你的复习笔记，因为它集中了该主题的所有核心公式。用新的例题逐条练习，将能从容应对 IGCSE 中的概率题目。

12. Practice Questions | 实战演练

Question 1: A fair die is rolled once. Find the probability that the result is a prime number.

问题 1： 掷一枚公平骰子一次，求结果是质数的概率。

The prime numbers on a die are 2, 3, and 5. Therefore P(prime) = 3/6 = 1/2.

骰子上的质数是 2、3 和 5。因此 P(质数) = 3/6 = 1/2。

Question 2: A bag contains 5 red and 3 white balls. Two balls are drawn at random without replacement. Find the probability that both balls are white.

问题 2： 一个袋子中有 5 个红球和 3 个白球。随机不放回地抽取两个球，求两个都是白球的概率。

P(white, white) = 3/8 × 2/7 = 6/56 = 3/28.

P(白, 白) = 3/8 × 2/7 = 6/56 = 3/28。

Question 3: The probability that a student passes mathematics is 0.7 and the probability that the same student passes physics is 0.6. If the events are independent, find the probability that the student passes both subjects.

问题 3： 某学生数学及格的概率为 0.7，物理及格的概率为 0.6。若两事件独立，求该学生两科都及格的概率。

P(pass both) = 0.7 × 0.6 = 0.42.

P(两科都及格) = 0.7 × 0.6 = 0.42。

Work through each problem on paper before checking your answer. Repetition builds fluency, and fluency leads to speed and accuracy in the examination.

请在纸上独立完成每一题后再核对答案。反复练习能提升熟练度，而熟练度能带来考试中的速度与准确率。

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Understanding Probability for IGCSE Mathematics | IGCSE 数学概率核心解析