GCSE WJEC Maths: Probability in Focus | GCSE WJEC 数学：概率考点精讲

📚 GCSE WJEC Maths: Probability in Focus | GCSE WJEC 数学：概率考点精讲

Probability is a core topic in the WJEC GCSE Mathematics specification, requiring you to understand chance, calculate probabilities for single and combined events, and apply rules for mutually exclusive and independent events. This revision guide walks you through the essential concepts, from the probability scale to tree diagrams, with bilingual explanations and worked examples to help you master every type of exam question.

概率是 WJEC GCSE 数学大纲中的核心内容，你需要理解随机现象、计算单一事件和组合事件的概率，并掌握互斥事件与独立事件的运算法则。这篇考点精讲会带你梳理从概率尺度到树形图的所有关键概念，双语讲解与典型例题将帮助你彻底吃透每一类考题。

1. What is Probability? | 什么是概率？

Probability describes how likely an event is to happen. It is a number between 0 and 1, often expressed as a fraction, decimal or percentage. A probability of 0 means the event is impossible; a probability of 1 means it is certain.

概率用于描述某个事件发生的可能性大小，它是一个介于 0 和 1 之间的数，通常用分数、小数或百分数表示。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。

The probability of an event A is written as P(A). The sum of the probabilities of all possible outcomes of an experiment must equal 1.

事件 A 的概率记作 P(A)。一次试验所有可能结果的概率之和必须等于 1。

P(Event) = Number of favourable outcomes / Total number of possible outcomes

P(事件) = 有利结果数 / 所有可能结果总数

2. The Probability Scale | 概率尺度

You should be able to mark probabilities on a number line from 0 to 1. Words such as ‘impossible’, ‘unlikely’, ‘even chance’, ‘likely’ and ‘certain’ correspond to positions on this scale.

你需要学会在 0 到 1 的数轴上标出概率。像 ‘不可能’、 ‘不太可能’、 ‘对等机会’、 ‘很可能’、 ‘必然’ 这些词语都对应着尺度上的固定位置。

0 – Impossible | 不可能
0.25 – Unlikely | 不太可能
0.5 – Even chance | 对等机会
0.75 – Likely | 很可能
1 – Certain | 必然

Being able to interpret real-life statements using the probability scale is a common non-calculator question.

用概率尺度解释现实生活中的表述，是常见的非计算器考题。

3. Equally Likely Outcomes | 等可能结果

When all outcomes are equally likely, probability is simply a ratio. For a fair six-sided dice, each number has a probability of 1/6. The probability of rolling an even number is 3/6 = 1/2.

当所有结果等可能时，概率就是一个简单的比值。对于一枚均匀的六面骰子，每个数字的概率都是 1/6。掷出偶数的概率是 3/6 = 1/2。

Watch out for bias. If a dice is ‘fair’, outcomes are equally likely; if it is ‘biased’, probabilities are not equal and must be given or calculated from relative frequency.

注意区分公平性与偏差。若骰子是 ‘公平的’，结果等可能；若是 ‘有偏的’，概率不相等，题目会给出或者需要通过相对频率来求。

4. Mutually Exclusive Events and the OR Rule | 互斥事件与加法法则

Two events are mutually exclusive if they cannot happen at the same time. For mutually exclusive events A and B, the probability that A or B occurs is the sum of their individual probabilities.

如果两个事件不能同时发生，它们就是互斥事件。对于互斥事件 A 和 B，A 或 B 发生的概率等于各自概率之和。

P(A or B) = P(A) + P(B) for mutually exclusive events

互斥事件：P(A 或 B) = P(A) + P(B)

If events are not mutually exclusive, you must subtract the overlap, but this is usually met at Higher tier only.

如果事件不是互斥的，需要减去重叠部分的概率，但这通常只在 Higher 级别才会涉及。

5. Tree Diagrams – Independent Events | 树形图 – 独立事件

Tree diagrams are powerful tools for showing all possible outcomes of two or more events. Each branch represents an outcome with its probability. For independent events, the probabilities on the second set of branches stay the same regardless of the first outcome.

树形图是展示两个或多个事件所有可能结果的强大工具。每条枝干代表一个结果及其概率。对于独立事件，第二层枝干的概率不会因为第一次的结果而改变。

To find the probability of a combination of outcomes, multiply the probabilities along the branches. Then add the probabilities of different branches if more than one combination satisfies the condition.

计算组合结果的概率时，将所经枝干上的概率相乘。如果有多种组合满足条件，再将各分支的概率相加。

Example: A bag has 3 red and 2 blue balls. Draw two balls with replacement. Probability of drawing at least one red can be found by considering paths: RR, RB, BR. P(at least one red) = (3/5 × 3/5) + (3/5 × 2/5) + (2/5 × 3/5) = 9/25 + 6/25 + 6/25 = 21/25.
例题：袋中有 3 红 2 蓝。有放回地抽两次。至少一个红的概率：考虑路径 RR, RB, BR。P(至少一红) = (3/5 × 3/5) + (3/5 × 2/5) + (2/5 × 3/5) = 21/25。

6. Tree Diagrams – Without Replacement | 树形图 – 不放回

When items are selected without replacement, the events are no longer independent because the fractions change after each pick. WJEC exams often test this at the Higher tier.

当物品被不放回地抽取时，事件就不再独立了，因为每次抽取后的分数会发生变化。WJEC 考试通常在 Higher 级别考查这一点。

You must update the probabilities on the second set of branches according to what has been removed. For instance, from a bag of 3 red and 2 blue balls, if you pick a red first and do not replace it, the bag now has 2 red and 2 blue, so probabilities become 2/4 for red and 2/4 for blue on the next pick.

你需要根据已移除的物品更新第二层枝干的概率。例如，从 3 红 2 蓝的袋中首次取出红球且不放回，袋中剩余 2 红 2 蓝，第二次红球概率为 2/4，蓝球为 2/4。

7. Conditional Probability – A First Look | 条件概率 – 初步认识

Conditional probability considers the chance of an event occurring given that another event has already happened. In WJEC GCSE, this often appears within tree diagrams. You might be asked to compute a probability like P(second red | first red).

条件概率指的是在已知另一事件已经发生的条件下某事件发生的概率。在 WJEC GCSE 中，这经常出现在树形图里。你可能需要计算诸如 P(第二次红 | 第一次红) 这样的概率。

Reading ‘given that’ statements carefully is crucial. The formula P(A|B) = P(A and B)/P(B) is usually introduced at A-level, but at GCSE you can reason directly from the diagram or a reduced sample space.

仔细审读 ‘已知……’ 的表述至关重要。P(A|B) = P(A 且 B)/P(B) 这个公式通常在 A-level 才引入，但 GCSE 阶段你可以直接从树形图或缩小的样本空间中推理得出。

8. Venn Diagrams and Probability | 维恩图与概率

Venn diagrams help visualise relationships between events, especially when grouping and overlapping occur. The rectangle represents the sample space, circles represent events, and overlapping regions show outcomes that belong to both events.

维恩图有助于把事件之间的关系形象化，尤其是在出现分组与重叠时。矩形代表样本空间，圆圈代表事件，重叠区域表示同时属于两个事件的结果。

A common task is to calculate probabilities from given numbers on a Venn diagram, or to complete missing numbers using the fact that P(A ∪ B) = P(A) + P(B) − P(A ∩ B) and that total probability is 1.

常见的任务是利用维恩图上给出的数字计算概率，或利用 P(A ∪ B) = P(A) + P(B) − P(A ∩ B) 以及总概率为 1 来补全缺失的数字。

9. Expected Frequency | 期望频率

If you know the probability of an event, you can predict how many times it is expected to occur in a number of trials. Expected frequency = probability × number of trials.

如果你知道某个事件的概率，就可以预测它在多次试验中预期出现的次数。期望频率 = 概率 × 试验次数。

Expected frequency = P(Event) × n

期望频率 = P(事件) × 试验次数

This concept is regularly examined. For example, a biased coin lands heads with probability 0.6. If you flip it 200 times, you would expect 0.6 × 200 = 120 heads.

这个知识点经常出现在考题中。例如，一枚有偏的硬币出现正面的概率为 0.6。若抛掷 200 次，则期望正面次数为 0.6 × 200 = 120。

10. Relative Frequency and Experimental Probability | 相对频率与实验概率

When a probability cannot be determined theoretically, you can estimate it using relative frequency from an experiment or survey. Relative frequency = number of successful trials / total number of trials.

当无法从理论上确定概率时，可以通过实验或调查中的相对频率来进行估计。相对频率 = 成功试验次数 / 总试验次数。

The more trials you carry out, the closer the relative frequency tends to get to the true theoretical probability. WJEC questions might ask you to compare experimental results with expected outcomes and comment on the differences.

进行的试验次数越多，相对频率就会越接近真实的理论概率。WJEC 的题目可能会要求你比较实验结果与期望结果，并对差异作出评论。

11. Using Outcomes and Sample Spaces | 列举结果与样本空间

For combined events like throwing two dice or spinning two spinners, writing out a systematic list or using a two-way table ensures you count all possibilities correctly. The sample space diagram is especially useful for questions involving sums or products.

对于掷两枚骰子或旋转两个转盘这样的组合事件，利用系统列表或双向表格能够确保你正确统计所有可能性。样本空间图在涉及和或积的题目中尤为实用。

Example: Spinner A has numbers 1, 2, 3; Spinner B has numbers 2, 4. A two-way table shows 3 × 2 = 6 equally likely outcomes. The probability of a sum greater than 4 can then be found by counting favourable pairs.

例如：转盘 A 有数字 1, 2, 3；转盘 B 有数字 2, 4。双向表显示 3 × 2 = 6 个等可能结果。然后可以通过数出和大于 4 的组合来求概率。

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

Always simplify fractions or give probabilities in the form requested. Check that your probabilities never exceed 1 or go below 0. When using tree diagrams, label branches clearly and multiply along the correct path.

永远要化简分数或按题目要求的形式给出概率。检查你的概率是否从不超过 1 或低于 0。使用树形图时，清楚地标注枝干，沿正确路径相乘。

Common mistake: confusing P(A or B) with P(A and B). For mutually exclusive events, use addition for ‘or’ and multiplication for ‘and’ only when events are independent. Read each question carefully to identify whether items are replaced or not.

常见错误：混淆 P(A 或 B) 与 P(A 且 B)。互斥事件用加法算 ‘或’，独立事件才能用乘法算 ‘且’。仔细读题，判断物品是否有放回。

Finally, always relate your answer back to the context, especially for expected frequency and relative frequency tasks where a written interpretation is often required.

最后，务必将答案结合具体情境来表述，尤其是在期望频率和相对频率的题目中，往往需要给出文字解释。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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GCSE WJEC Maths: Probability in Focus | GCSE WJEC 数学：概率 考点精讲