📚 013 Probability | 013 概率
Probability is the mathematics of chance, allowing us to quantify uncertainty and make informed predictions about the outcomes of experiments. In IGCSE Edexcel Mathematics, you will learn how to calculate probabilities for single and combined events, use tree diagrams and Venn diagrams, and apply the rules of addition and multiplication for mutually exclusive and independent events. Mastering this topic is essential not only for your exam success but also for understanding statistics, risk, and decision-making in real life.
概率是研究随机现象的数学分支，帮助我们量化不确定性并对试验结果做出合理预测。在IGCSE Edexcel数学课程中，你将学习如何计算单一事件和复合事件的概率，使用树形图和维恩图，以及应用互斥事件和独立事件的加法与乘法法则。掌握这一主题不仅对考试成功至关重要，也是理解统计、风险及现实决策的基础。
1. What Is Probability? | 什么是概率？
Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1. An event with probability 0 is impossible, while an event with probability 1 is certain. You can also write probabilities as fractions, decimals, or percentages. For example, the probability of flipping a fair coin and getting heads is 0.5, ½, or 50%.
概率是度量事件发生可能性的数值，范围在0到1之间。概率为0表示不可能事件，概率为1表示必然事件。概率可以用分数、小数或百分数表示。例如，抛一枚均匀硬币得到正面的概率是0.5、½或50%。
In probability notation, P(A) denotes the probability of event A occurring. The formula for theoretical probability is:
在概率符号中，P(A) 表示事件 A 发生的概率。理论概率的公式为：
P(A) = Number of favourable outcomes ÷ Total number of possible outcomes
P(A) = 有利结果数 ÷ 所有可能结果总数
2. Sample Spaces and Events | 样本空间与事件
outcomes. An event is any subset of the sample space. For example, when rolling a fair six‑sided die, the sample space is S = {1, 2, 3, 4, 5, 6}. The event “rolling an even number” is {2, 4, 6}. The probability of rolling an even number is 3/6 = 1/2.
试验的样本空间 (S) 是所有可能结果的集合。事件是样本空间的任意子集。例如，掷一枚均匀六面骰子时，样本空间 S = {1, 2, 3, 4, 5, 6}。事件“掷出偶数点”为 {2, 4, 6}，其概率为 3/6 = 1/2。
You can list the sample space systematically using tables, especially for two‑part experiments. A two‑way table displays all combinations of two events, making it easy to count outcomes and find probabilities.
你可以通过表格系统列出样本空间，尤其是在两阶段试验中。双向表格展示两个事件的所有组合，便于计数结果和求概率。
3. The Probability Scale | 概率尺度
The probability scale places events on a line from 0 to 1. Words such as ‘impossible’, ‘unlikely’, ‘even chance’, ‘likely’, and ‘certain’ describe positions along the scale. Understanding this scale helps you check whether your calculated probabilities are reasonable.
概率尺度将事件标记在0到1的线段上。用“不可能”、“不太可能”、“等可能”、“很可能”和“必然”等词语描述尺度上的位置。理解这个尺度有助于检查计算出的概率是否合理。
ility that a randomly chosen person was born on a Monday is 1/7 ≈ 0.14, which falls in the ‘unlikely’ region, while the probability that a new‑born baby is a girl is about 0.48, close to an even chance.
例如，随机选一个人，其出生在星期一的概率为 1/7 ≈ 0.14，属于“不太可能”区域；而新生婴儿是女孩的概率约为 0.48，接近等可能。
4. Relative Frequency vs Theoretical Probability | 相对频率与理论概率
Theoretical probability is what you expect to happen mathematically, based on equally likely outcomes. Relative frequency (or experimental probability) is what actually happens when you perform an experiment multiple times, calculated as:
理论概率是基于等可能结果的数学预期。相对频率（或实验概率）是多次试验中实际发生的结果，计算公式为：
Relative frequency = Number of times the 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. This is known as the Law of Large Numbers.
随着试验次数增加，相对频率会趋向于理论概率，这被称为大数定律。
In IGCSE questions, you may be asked to compare relative frequencies with theoretical probabilities or to estimate probabilities from experimental data where outcomes are not equally likely.
在IGCSE考题中，可能会要求你比较相对频率与理论概率，或根据非等可能结果的实验数据估计概率。
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, when rolling a die, getting a 2 and getting a 5 are mutually exclusive. For mutually exclusive events A and B, the probability of A or B occurring is the sum of their individual probabilities:
如果两个事件不可能同时发生，则它们是互斥的。例如，掷骰子得到2和得到5互斥。对于互斥事件 A 和 B，A 或 B 发生的概率是各自概率之和：
P(A or B) = P(A) + P(B)
If events are not mutually exclusive—that is, they can both occur together—you must subtract the overlap to avoid double counting:
如果事件不是互斥的，即它们可以同时发生，则必须减去重叠部分以避免重复计算：
P(A or B) = P(A) + P(B) − P(A and B)
Common exam questions involve selecting a card at random from a deck or a student from a class and applying the correct addition rule.
常见考题包括从一副纸牌中随机抽牌或从一个班级中随机选择学生，并应用正确的加法法则。
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 example, flipping a coin and rolling a die are independent. For independent events A and B, the probability of both A and B occurring is the product of their probabilities:
如果一件事的发生不影响另一件事的概率，则这两个事件相互独立。例如，抛硬币和掷骰子相互独立。对于相互独立的事件 A 和 B，两者同时发生的概率是各自概率的乘积：
P(A and B) = P(A) × P(B)
It is crucial to check whether events are independent or dependent before multiplying. If events are dependent, you must use conditional probability or tree diagrams, adjusting the second probability based on the outcome of the first.
在相乘之前，检查事件是否独立至关重要。如果事件是相关的，则必须使用条件概率或树形图，根据第一个事件的结果调整第二个事件的概率。
7. Tree Diagrams | 树形图
Tree diagrams are a powerful visual tool for handling two or more stages of an experiment, especially when events are dependent (non‑replacement problems). Each branch represents a possible outcome, labelled with its probability. Multiply probabilities along branches to find the probability of a combined outcome, and add branch probabilities for alternative routes to the same final event.
树形图是处理两阶段或多阶段试验的强大可视化工具，特别适用于事件相关的情况（不放回问题）。每条分枝代表一个可能结果，并标注其概率。沿分枝相乘可得到组合结果的概率；如果同一最终事件有多条路径，则将相关分枝的概率相加。
3/10. A tree diagram clearly shows how the denominator changes because the first ball is not replaced.
例如，袋子中有3个红球和2个蓝球。如果不放回地依次抽取两球，两个都是红球的概率为 (3/5) × (2/4) = 6/20 = 3/10。树形图清楚地展示了由于不放回，分母如何变化。
Always label branches with their probabilities, write final outcomes at the end, and check that probabilities on branches from any point sum to 1. In IGCSE exams, tree diagrams are frequently worth several marks, so clarity is key.
务必在分枝上标注概率，在末端写出最终结果，并确保从同一点出发的各分枝概率之和为1。在IGCSE考试中，树形图通常占好几分，因此表述清晰至关重要。
8. Conditional Probability | 条件概率
Conditional probability deals with the probability of an event given that another event has already occurred. It is denoted by P(A|B), read as “the probability of A given B”. The formula is:
条件概率研究在另一事件已经发生的条件下，某个事件发生的概率。记作 P(A|B)，读作“在B发生的条件下A的概率”。公式为：
P(A|B) = P(A and B) ÷ P(B)
This concept is heavily tested in IGCSE, often in the context of two‑way tables or Venn diagrams. For instance, you may be asked to find the probability that a student studies French given that they study Spanish.
这一概念在IGCSE中是重点考查内容，常以双向表格或维恩图为背景。例如，你可能会被问到已知一名学生学习西班牙语，求其学习法语的概率。
Conditional probability is intimately connected with dependent events in tree diagrams: when you write a probability on the second set of branches, you are actually using conditional probabilities. Understanding this link helps you avoid common errors.
条件概率与树形图中的相关事件紧密相连：当你在第二层分枝上写下概率时，实际上就是在使用条件概率。理解这一联系有助于避免常见错误。
9. Venn Diagrams and Probability | 维恩图与概率
Venn diagrams show sets and their intersections inside a rectangle that represents the universal set. You can use Venn diagrams to calculate probabilities involving “and”, “or”, and “not” by considering regions. For two sets A and B, the intersection P(A ∩ B) and union P(A ∪ B) can be easily visualised.
维恩图在表示全集的长方形内展示集合及其交集。通过考虑各个区域，你可以用维恩图计算涉及“且”、“或”和“非”的概率。对于两个集合A和B，交集 P(A ∩ B) 和并集 P(A ∪ B) 可以直观呈现。
To find the probability that a randomly chosen element belongs to a particular region, divide the number of elements in that region by the total number of elements in the universal set. Always remember to account for elements outside both sets.
要计算随机选取的元素属于某个特定区域的概率，将该区域的元素个数除以全集中元素的总数即可。务必记得考虑两个集合外部的元素。
Venn diagrams are especially useful for solving problems where you are given overlapping data, such as students taking Science and Art. A clear diagram can make the probability calculation straightforward.
维恩图在解决具有重叠数据的问题时特别有用，例如同时修读科学和艺术的学生。一个清晰的图形可以使概率计算变得简单明了。
10. Expected Frequency | 期望频数
The expected frequency is the number of times you would expect an event to occur in a given number of trials. It is calculated by multiplying the probability of the event by the number of trials:
期望频数是指在给定试验次数中，预期某事件发生的次数。计算方法为事件概率乘以试验次数：
Expected frequency = P(A) × Number of trials
期望频数 = P(A) × 试验次数
00 = 6. This does not mean exactly 6 will be faulty, but it is the average over many days.
例如，如果一个灯泡有瑕疵的概率是0.02，某商店每天卖出300个灯泡，那么有瑕疵灯泡的期望数量为 0.02 × 300 = 6。这并不是说恰好会有6个瑕疵品，而是长期的平均值。
Expected frequency questions often link probability with real‑life contexts and are a source of easy marks if you can correctly identify the probability and apply the multiplication.
期望频数问题常将概率与实际情境联系起来，如果你能正确识别概率并应用乘法，这就是容易得分的部分。
11. Exam Tips and Common Mistakes | 考试技巧与常见错误
IGCSE Edexcel probability questions test both straightforward calculations and deeper reasoning. Pay close attention to the wording: “or” usually requires addition, “and” usually multiplication, but only after verifying independence or mutual exclusivity. Always check whether the situation involves replacement or not, as this changes probabilities on subsequent branches.
IGCSE Edexcel的概率题既考查直接计算，也考查深层推理。要仔细审题：“或”通常需要加法，“且”通常用乘法，但必须先验证独立性或互斥性。务必检查情境中是否涉及放回，因为它会改变后续分枝的概率。
Common mistakes include forgetting to adjust denominators in non‑replacement problems, adding fractions without a common denominator, confusing P(A or B) with P(A and B), and failing to label tree diagram branches clearly. Always write your working step by step, showing the calculation and simplification of fractions.
常见错误包括在不放回问题中忘记调整分母、分数加法时未通分、混淆 P(A或B) 与 P(A且B)，以及未清晰标注树形图的分枝。务必逐步展示计算过程，并化简分数。
When answering probability questions, make sure your final probability is in its simplest form unless the question states otherwise. Also, if you are asked to describe something as “impossible” or “certain”, use the exact words from the probability scale.
回答概率题时，除非题目另有要求，否则应将最终概率化为最简形式。此外，如果要求用“不可能”或“必然”描述某事件，请使用概率尺度上的准确用语。
12. Summary | 知识总结
total, addition rule for mutually exclusive events, multiplication rule for independent events, and the conditional probability formula. Tree diagrams and Venn diagrams give you a structured way to manage complicated multi‑stage and overlapping events.
IGCSE Edexcel数学中的概率综合了分数运算、逻辑推理和可视化工具。记住核心公式：P(A) = 有利结果 / 总数；互斥事件的加法法则；独立事件的乘法法则以及条件概率公式。树形图和维恩图为你提供了处理复杂多阶段和重叠事件的结构化方法。
| Key Concept | Formula / Rule |
| Basic Probability | P(A) = favourable / total |
| Mutually Exclusive Events | P(A or B) = P(A) + P(B) |
| Non‑Mutually Exclusive | P(A or B) = P(A) + P(B) − P(A and B) |
| Independent Events | P(A and B) = P(A) × P(B) |
| Conditional Probability | P(A|B) = P(A and B) / P(B) |
| Expected Frequency | Expected = P(A) × number of trials |
With regular practice on past paper questions, these concepts will become second nature, and you will be able to tackle even the trickiest IGCSE probability problems with confidence.
通过定期练习往年试卷，这些概念将变得得心应手，你就能自信地解决IGCSE中最为棘手的概率问题。
Published by TutorHao | IGCSE Edexcel Mathematics Revision Series | aleveler.com
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