📚 IGCSE Maths: Probability – Key Concepts Review | IGCSE 数学：概率 考点精讲
Probability in IGCSE Mathematics is a core topic that builds your understanding of chance, uncertainty, and prediction. It links theoretical knowledge with real-world applications, from weather forecasts to game strategies. Mastering probability requires clarity on key terms, the ability to calculate probabilities using fractions, decimals, or percentages, and the skill to apply rules for combined events. This guide systematically covers everything you need, from basic definitions to tree diagrams and conditional probability, following the Cambridge IGCSE syllabus (both Core and Extended).
IGCSE 数学中的概率是一个核心章节，帮助你理解随机性、不确定性与预测。它将理论知识与天气预报、游戏策略等实际应用紧密相连。要掌握概率，你需要清晰理解基本术语，能熟练用分数、小数或百分比计算概率，并能灵活运用组合事件的运算规则。这篇精讲系统梳理了从基本定义到树形图、条件概率的全部考点，覆盖剑桥 IGCSE 大纲的 Core 与 Extended 内容。
1. Basic Probability Concepts | 基本概率概念
Probability measures how likely an event is to happen. It can be expressed as a fraction, decimal, or percentage. The probability scale ranges from 0 (impossible) to 1 (certain). All other probabilities lie strictly between these two extremes.
概率度量某个事件发生的可能性大小，可用分数、小数或百分数表示。概率尺度从 0（不可能）到 1（必然），其余所有概率都严格落在这两点之间。
An experiment is any process that produces an outcome, such as tossing a coin. The sample space is the set of all possible outcomes. An event is a particular subset of the sample space that we are interested in. For example, when rolling a fair six‑sided die, the sample space is {1, 2, 3, 4, 5, 6}. The event ‘rolling an even number’ is {2, 4, 6}.
进行一次操作并产生结果的过程称为试验，比如抛硬币。样本空间是所有可能结果的集合。事件则是样本空间中我们感兴趣的一个子集。例如，掷一枚均匀的六面骰子，样本空间是 {1, 2, 3, 4, 5, 6}，事件“掷出偶数”就是 {2, 4, 6}。
Equally likely outcomes are outcomes that have the same chance of occurring, like the faces of a fair die. When outcomes are not equally likely, probabilities cannot simply be found by counting, and we often need experimental data or given weights.
等可能结果是指发生机会相同的结果，比如公平骰子的每一个面。当结果不是等可能时，就不能仅靠计数求概率，通常需要实验数据或给定的加权信息。
2. Calculating Theoretical Probability | 理论概率计算
If all outcomes in a sample space are equally likely, the probability of an event E is given by: P(E) = (number of outcomes in E) ÷ (total number of outcomes in the sample space). This is the fundamental formula that you will use repeatedly.
如果样本空间中所有结果等可能，事件 E 的概率由公式给出：P(E) = E 中的结果数 ÷ 样本空间的总结果数。这是你将反复使用的基本公式。
For example, a bag contains 3 red, 2 blue, and 5 green counters. The probability of picking a blue counter at random is 2 / (3+2+5) = 2/10 = 1/5. Remember to always simplify fractions unless the question specifies otherwise.
例如，一个袋子里有 3 个红色、2 个蓝色和 5 个绿色筹码。随机取出一个蓝色筹码的概率是 2 / (3+2+5) = 2/10 = 1/5。记得除非题目另有要求，否则总要将分数化为最简形式。
Probability can also be written as a decimal or percentage. 1/5 = 0.2 = 20%. In IGCSE exams, you may be asked to give your answer in a specific form – read the instruction carefully.
概率也可以用小数或百分数表示，1/5 = 0.2 = 20%。在 IGCSE 考试中，题目可能会指定答案的形式，务必仔细阅读要求。
When listing outcomes, systematic methods like ordered lists or two‑way tables prevent missing any possibilities. This is especially helpful in questions involving two dice or spinners.
列举结果时，采用有序列表或二维表格等系统化方法可以避免遗漏。在涉及两粒骰子或两个转盘的题目中，这一点格外有用。
3. Probability Scale and Notation | 概率尺度与记号
Probabilities are always numbers between 0 and 1 inclusive. You should be comfortable marking probabilities on a number line and interpreting words such as ‘likely’, ‘unlikely’, ‘even chance’, ‘certain’, and ‘impossible’.
概率始终是 0 到 1 之间（含两端）的数字。你需要能在数轴上标出概率的位置，并理解“很可能”“不太可能”“机会均等”“必然”“不可能”等词语的含义。
We use the notation P(A) to represent ‘the probability of event A’. The complement of event A, written as A’ or not A, has probability P(A’) = 1 – P(A). This is valuable when it is easier to calculate the chance of something not happening.
我们用 P(A) 表示“事件 A 的概率”。事件 A 的补集，记作 A’ 或“不是 A”，其概率为 P(A’) = 1 – P(A)。当反向计算更容易时，这个关系非常有用。
For instance, if the probability of rain tomorrow is 0.3, then the probability of no rain is 1 – 0.3 = 0.7. Always check that your answer makes sense within the 0–1 scale.
例如，若明天下雨的概率是 0.3，那么不下雨的概率就是 1 – 0.3 = 0.7。永远要检查答案是否落在 0–1 的合理范围内。
4. Experimental Probability and Relative Frequency | 实验概率与相对频率
Experimental probability, or relative frequency, is based on actual trials rather than theory. It is calculated as: relative frequency = (number of times the event occurs) ÷ (total number of trials).
实验概率，也称相对频率，是基于实际试验而非理论推导的。它的计算公式为：相对频率 = 事件发生的次数 ÷ 试验总次数。
If you toss a coin 100 times and obtain 48 heads, the experimental probability of heads is 48/100 = 0.48. As the number of trials increases, the relative frequency tends to get closer to the theoretical probability – this is known as the law of large numbers.
如果你抛一枚硬币 100 次得到 48 次正面，那么出现正面的实验概率是 48/100 = 0.48。随着试验次数增加，相对频率会逐渐趋近理论概率，这就是大数定律。
IGCSE questions often ask you to compare experimental and theoretical values, or to estimate the number of trials from given frequencies. Always base your estimate on the proportion observed.
IGCSE 考题经常要求你比较实验概率与理论值，或者根据给定的频率估计试验次数。始终基于观测到的比例进行估算。
5. Expectation (Expected Frequency) | 期望频率
Expectation predicts how many times an event will occur in a fixed number of trials, assuming the probability remains constant. Expected frequency = probability of the event × number of trials.
期望用于预测在固定试验次数下事件将发生的次数，前提是概率保持不变。期望频率 = 事件的概率 × 试验次数。
For example, a biased spinner lands on red with probability 0.25. If you spin it 200 times, the expected number of reds is 0.25 × 200 = 50. Note that the actual outcome may differ; expectation is not a guarantee.
例如，一个偏斜转盘停在红色区域的概率是 0.25。转动 200 次，期望红色出现的次数是 0.25 × 200 = 50。注意实际结果可能不同，期望并不是保证。
This concept appears frequently in probability questions involving large numbers of trials, and is an excellent way to check whether your probability calculation is plausible.
这个概念常出现在涉及大量试验的概率题中，也是检验你的概率计算是否合理的好方法。
6. Mutually Exclusive Events and Addition Rule | 互斥事件与加法法则
Two events are mutually exclusive if they cannot occur at the same time. For any mutually exclusive events A and B, the addition rule applies: P(A or B) = P(A) + P(B).
如果两个事件不能同时发生，则称它们为互斥事件。对于任意互斥事件 A 和 B，适用加法法则：P(A 或 B) = P(A) + P(B)。
A common example is drawing a single card: the events ‘drawing a King’ and ‘drawing a Queen’ are mutually exclusive. If a bag contains only red, blue, and green balls, picking a red and picking a blue are mutually exclusive.
一个常见的例子是抽一张牌：“抽到 K”与“抽到 Q”这两个事件互斥。如果袋子里只有红、蓝、绿三种球，那么摸到红球和摸到蓝球就是互斥的。
For events that are not mutually exclusive, you must subtract the overlap: P(A or B) = P(A) + P(B) – P(A and B). This is the general addition rule, often used with Venn diagrams.
对于非互斥事件，必须减去重叠部分：P(A 或 B) = P(A) + P(B) – P(A 且 B)。这是通用的加法法则，常搭配维恩图使用。
7. Independent Events and Multiplication Rule | 独立事件与乘法法则
Two events are independent if the outcome of one does not affect the probability of the other. For independent events A and B, the multiplication rule holds: P(A and B) = P(A) × P(B).
如果两个事件互不影响，则它们是独立事件。对于独立事件 A 和 B，乘法法则成立：P(A 且 B) = P(A) × P(B)。
For example, rolling a fair die and flipping a fair coin are independent. The probability of rolling a 4 and getting heads is 1/6 × 1/2 = 1/12. Dependence occurs when you do not replace an item, such as picking sweets from a bag without replacement.
例如，掷一枚均匀骰子并抛一枚均匀硬币是独立的。掷出 4 且得到正面的概率是 1/6 × 1/2 = 1/12。当你从一个袋子里取出物品且不放回时，事件之间就产生了依赖关系。
Questions often require you to decide whether events are independent. Look for the phrases ‘with replacement’ (independent) and ‘without replacement’ (dependent).
题目经常要求你判断事件是否独立。注意“放回”（独立）与“不放回”（不独立）这类关键表述。
8. Tree Diagrams | 树形图
Tree diagrams are powerful tools for showing all possible outcomes of two or more events, especially when events are combined and probability values change (e.g., without replacement). Each branch is labeled with its probability, and the final outcome probabilities are found by multiplying along the branches.
树形图是展示两个或多个事件所有可能结果的强大工具，特别适用于事件组合且概率值会改变的情形（例如不放回抽取）。每条分支标注其概率，最终结果的概率通过沿分支相乘得到。
For a bag with 3 red and 2 blue counters, picking two counters without replacement can be shown on a tree. The first pick has probabilities 3/5 (R) and 2/5 (B). The second pick probabilities change depending on the first outcome. The probability of RR is (3/5) × (2/4) = 6/20 = 3/10.
一个装有 3 个红球和 2 个蓝球的袋子，不放回地摸出两个球，可以用树形图表示。第一次摸出的概率分别是 3/5 (R) 和 2/5 (B)，第二次的概率则根据第一次结果而改变。RR 的概率为 (3/5) × (2/4) = 6/20 = 3/10。
When constructing tree diagrams, always check that the probabilities on branches from the same point sum to 1. The sum of all final outcome probabilities must also be 1 – a useful check.
构建树形图时，一定要检查同一点发出的各分支概率之和为 1。所有最终结果的概率之和也应为 1，这是有效的验证手段。
9. Conditional Probability (Extended) | 条件概率（扩展）
Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted by P(A|B) and calculated as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.
条件概率是指在另一个事件已经发生的条件下，某一事件发生的概率。记作 P(A|B)，计算公式为 P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。
You might encounter this directly in shading regions on Venn diagrams or in given formula questions. Tree diagram probabilities for the second event are actually conditional probabilities, especially in without‑replacement scenarios.
你可能会在维恩图涂色区域或给定公式的题目中直接遇到条件概率。树形图中第二次事件的概率其实就是条件概率，尤其在不放回的情境下。
For example, if P(A) = 0.6, P(B) = 0.4, and P(A ∩ B) = 0.2, then P(A|B) = 0.2 / 0.4 = 0.5. Interpret this as: among the times B occurs, A occurs half of the time.
举例来说，若 P(A) = 0.6，P(B) = 0.4，P(A ∩ B) = 0.2，则 P(A|B) = 0.2 / 0.4 = 0.5。这可以解读为：在 B 发生的那些情形中，A 发生了一半的时间。
10. Venn Diagrams and Probability | 维恩图与概率
Venn diagrams show sets and their intersections using overlapping circles. They are excellent for organising information about events that are not mutually exclusive, and for calculating probabilities involving ‘and’, ‘or’, and ‘not’.
维恩图利用交叠的圆来表示集合及其交集。它们特别适合整理非互斥事件的信息，以及计算涉及“且”“或”“非”的概率。
When a Venn diagram is drawn, the probability of an event is the sum of the probabilities of the individual regions making up that event. Always label each region with its probability or frequency and start from the intersection when the overall totals are known.
绘制维恩图后，某事件的概率就是组成该事件各区域概率的总和。务必在每个区域标注概率或频数，并且在已知整体总数时，从交集入手填充信息。
For two events A and B, the probability of ‘A or B’ is the sum of probabilities in all parts of the diagram except the outside. This visually demonstrates P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
对于两个事件 A 和 B，“A 或 B”的概率就是图中除了外部区域外所有部分的概率之和，这直观地展示了 P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
11. Systematic Listing and Sample Spaces | 系统列举与样本空间
For many IGCSE problems, especially those with two dice or two spinners, listing the sample space systematically prevents errors. A two‑way table is the most common method: list outcomes of the first event along the top and the second event down the side; then fill in all combined outcomes.
在许多 IGCSE 问题中，特别是涉及两粒骰子或两个转盘时，系统化列举样本空间可避免出错。最常用的方法是双向表格：上方列出第一个事件的结果，左侧列出第二个事件的结果，然后填充所有组合结果。
When two unbiased dice are rolled, there are 36 equally likely ordered pairs. The probability of scoring a sum of 7 is 6/36 = 1/6, because the favourable pairs are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Systematic listing makes this clear.
掷两粒均匀骰子时，共有 36 种等可能的有序数对。得到点数之和为 7 的概率是 6/36 = 1/6，因为有利数对包括 (1,6)、(2,5)、(3,4)、(4,3)、(5,2)、(6,1)。系统列举使之一目了然。
You can also use lists in braces or probability space diagrams. The key is to be organised – a chaotic list is a recipe for missing outcomes.
你也可以使用花括号内的列表或概率空间图。关键在于条理清晰——杂乱无章的列举极易遗漏结果。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
One of the most frequent errors is adding probabilities without checking whether events are mutually exclusive. Always ask: could these two events happen together? If yes, use the general addition rule with the intersection.
最常见的错误之一是未检查事件是否互斥就直接相加概率。永远先问自己：这两个事件能同时发生吗？如果能，就必须使用包含交集的通用加法法则。
Another mistake is confusing independent with mutually exclusive. Independence means one event does not affect the other’s probability; mutual exclusivity means they cannot both occur. These are different concepts and must not be swapped.
另一个错误是混淆独立事件与互斥事件。独立意味着一个事件的发生不影响另一个的概率；互斥则意味着两者不可能同时发生。这是截然不同的概念，绝不能互换使用。
In tree diagrams, ensure you update probabilities correctly after each intermediate event, especially in without‑replacement problems. Counters, sweets, or balls are commonly used; always reduce the total accordingly.
在树形图中，尤其是在不放回问题中，要确保每次中间事件后都正确更新概率。筹码、糖果或小球是常见物品——记得相应减少总数。
Finally, present your answers clearly as simplified fractions, decimals, or percentages as instructed. Always re‑read the question to ensure your interpretation of ‘or’, ‘and’, ‘at least’, ‘exactly’ is correct.
最后，按题目要求清晰给出答案，用化简的分数、小数或百分数呈现。务必再次审题，确保对“或”“且”“至少”“恰好”的理解准确无误。
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