📚 IGCSE AQA Maths: Probability Key Points | IGCSE AQA 数学：概率 考点精讲
Probability is a fundamental topic in the IGCSE AQA Mathematics specification, testing your ability to calculate chances, analyse outcomes, and apply logical reasoning to real-world situations. This article breaks down all the essential concepts you need to master, from basic probability rules to tree diagrams, conditional probability, and Venn diagrams, ensuring you are fully prepared for your exams.
概率是 IGCSE AQA 数学大纲中的基础主题，考查你计算机会、分析结果并将逻辑推理应用于现实情境的能力。本文分解了你需要掌握的所有核心概念——从基本概率规则到树状图、条件概率和维恩图，确保你为考试做好充分准备。
1. Probability Basics | 概率基础
Probability is a measure of how likely an event is to happen, ranging from 0 (impossible) to 1 (certain). It can be expressed as a fraction, decimal, or percentage.
概率是衡量事件发生可能性大小的度量，范围从 0（不可能）到 1（必然）。它可以用分数、小数或百分比表示。
For any event A, the probability is: P(A) = Number of favourable outcomes / Total number of equally likely outcomes.
对于任意事件 A，概率为：P(A) = 有利结果数目 / 等可能结果总数。
= 0; if it is certain, P(certain) = 1. All probabilities lie in the interval 0 ≤ P(A) ≤ 1.
如果事件不可能发生，P(不可能) = 0；如果事件必然发生，P(必然) = 1。所有概率都满足 0 ≤ P(A) ≤ 1。
2. Sample Space | 样本空间
The sample space is the set of all possible outcomes of an experiment. It can be listed, shown in a table, or illustrated with a diagram.
样本空间是一个实验所有可能结果的集合。它可以列出、用表格表示或用图表展示。
For example, when rolling a fair six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. Tossing a coin twice gives the sample space {HH, HT, TH, TT}.
例如，抛掷一枚均匀的六面骰子，样本空间为 {1, 2, 3, 4, 5, 6}。抛掷一枚硬币两次，样本空间为 {正正, 正反, 反正, 反反}。
Using a sample space diagram helps to visualise combined events and ensure no outcomes are missed.
使用样本空间图有助于可视化组合事件，并确保不遗漏任何结果。
3. Calculating Probabilities | 计算概率
Probabilities of events are calculated by counting favourable outcomes and dividing by the total number of outcomes, assuming all outcomes are equally likely.
假设所有结果等可能，事件的概率通过计算有利结果数目除以结果总数得到。
, where n(A) is the number of outcomes in A and n(S) is the total number in the sample space.
对于单个事件 A，P(A) = n(A) / n(S)，其中 n(A) 是事件 A 的结果数，n(S) 是样本空间总数。
For combined events, a systematic approach, such as a sample space table or tree diagram, is often used.
对于组合事件，通常使用系统方法，如样本空间表格或树状图。
4. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. For such events, the addition rule simplifies.
如果两个事件不能同时发生，则它们互斥。对于此类事件，加法法则简化为。
For mutually exclusive events A and B: P(A or B) = P(A) + P(B).
对于互斥事件 A 和 B：P(A 或 B) = P(A) + P(B)。
, the probability of getting a King or a Queen is P(King) + P(Queen) = 4/52 + 4/52 = 8/52.
示例：从一副牌中抽一张牌，得到 K 或 Q 的概率为 P(K) + P(Q) = 4/52 + 4/52 = 8/52。
t mutually exclusive, you must subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).
如果事件不互斥，必须减去重叠部分：P(A 或 B) = P(A) + P(B) − P(A 且 B)。
5. Independent Events | 独立事件
Independent events are those where the outcome of one does not affect the outcome of the other. The key multiplication rule applies.
独立事件是指一个事件的结果不影响另一个事件结果的事件。关键乘法法则适用。
For independent events A and B: P(A and B) = P(A) × P(B).
对于独立事件 A 和 B：P(A 且 B) = P(A) × P(B)。
Example: flipping a fair coin and rolling a die. P(heads and a 5) = 1/2 × 1/6 = 1/12.
示例：抛一枚均匀硬币并掷骰子。P(正面且 5 点) = 1/2 × 1/6 = 1/12。
This rule must be used with caution; always check whether events are truly independent before multiplying probabilities.
必须谨慎使用该规则；在相乘概率之前，务必检查事件是否真的独立。
6. Expected Frequency | 期望频数
The expected frequency of an event is the number of times we expect that event to occur in a repeated experiment.
期望频数是指在重复实验中我们期望事件发生的次数。
Expected frequency = probability × number of trials.
期望频数 = 概率 × 试验次数。
in on a given day is 0.3, the expected number of rainy days in 100 days is 0.3 × 100 = 30.
如果某天下雨的概率为 0.3，那么 100 天中降雨的期望天数为 0.3 × 100 = 30。
This concept links theoretical probability to real-world predictions and can be tested with relative frequency.
这个概念将理论概率与现实预测联系起来，并且可以通过相对频率进行检验。
7. Relative Frequency | 相对频率
Relative frequency is an estimate of probability based on experiments or historical data. It is calculated as:
相对频率是基于实验或历史数据对概率的估计。其计算方法为：
Relative frequency = (Number of times event occurs) / (Total number of trials).
相对频率 = (事件发生次数) / (试验总次数)。
As the number of trials increases, the relative frequency tends to get closer to the theoretical probability, though not exactly.
随着试验次数的增加，相对频率往往趋近于理论概率，但不会完全相等。
Exam questions may ask you to work with experimental data and use relative frequency as a probability estimate for future events.
考试题目可能要求你处理实验数据，并利用相对频率作为未来事件的概率估计。
8. Tree Diagrams | 树状图
Tree diagrams are used to display all possible outcomes of a sequence of events, showing the probabilities on each branch.
树状图用于展示一系列事件的所有可能结果，并在每条分支上标出概率。
At each stage, the sum of probabilities from a single point is 1. To find the probability of a combined path, multiply along the branches.
在每一阶段，从同一点出发的各分支概率之和为 1。要计算组合路径的概率，沿分支相乘。
For two independent events, the outcome probabilities are independent; for dependent events (without replacement), probabilities on the second branches change.
对于两个独立事件，结果概率独立；对于非独立事件（无放回），第二阶段分支的概率会改变。
Example with replacement: bag has 3 red, 7 blue. Pick, replace, pick again. P(both red) = 3/10 × 3/10 = 9/100.
有放回示例：袋中有 3 红 7 蓝。抽取后放回，再抽一次。P(两次都红) = 3/10 × 3/10 = 9/100。
Without replacement: same bag, pick 2 without replacement. P(both red) = 3/10 × 2/9 = 6/90 = 1/15.
无放回：同一袋子，不放回地抽 2 个。P(两个都红) = 3/10 × 2/9 = 6/90 = 1/15。
9. Conditional Probability | 条件概率
Conditional probability is the probability of event A occurring given that event B has already occurred. Notation: P(A | B).
条件概率是在事件 B 已经发生的条件下事件 A 发生的概率。记作 P(A | B)。
Formula: P(A | B) = P(A and B) / P(B), provided P(B) > 0.
公式：P(A | B) = P(A 且 B) / P(B)，前提是 P(B) > 0。
Conditional probability often appears in tree diagram problems and in questions involving 'given that' statements.
条件概率常出现在树状图问题和包含 '已知...' 陈述的题目中。
Example: from a deck, P(King | Face card) = P(King and Face card) / P(Face card) = (4/52) / (12/52) = 4/12 = 1/3.
示例：从一副牌中，P(K | 人头牌) = P(K 且人头牌
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