GCSE Maths: Probability Revision Guide | GCSE 数学：概率考点精讲

📚 GCSE Maths: Probability Revision Guide | GCSE 数学：概率考点精讲

Probability is the measure of how likely an event is to happen. In GCSE Maths, you are expected to understand the probability scale, calculate theoretical and experimental probabilities, use diagrams such as sample spaces, Venn diagrams and tree diagrams, and work with combined events. Whether you are sitting Foundation or Higher tier, probability is a key topic that tests your logical thinking and your ability to interpret real-world situations mathematically.

概率是衡量事件发生可能性大小的度量。在 GCSE 数学中，你需要理解概率尺度，计算理论概率与实验概率，使用样本空间、韦恩图和树形图等图表，以及处理组合事件。无论你参加的是基础层级还是高级层级考试，概率都是一个考查逻辑思维与用数学解读现实情境能力的重要板块。

1. The Probability Scale | 概率尺度

All probabilities lie between 0 and 1, inclusive. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain. You can also express probabilities as fractions, decimals or percentages.

所有概率值都介于 0 和 1 之间（包括 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。你也可以用分数、小数或百分数来表示概率。

Words such as ‘likely’, ‘unlikely’, ‘evens’ and ‘certain’ can be placed on the probability scale. For example, an event with probability 0.6 is more likely than not, while an event with probability 0.1 is very unlikely.

“很可能”“不大可能”“对半开”“必然”等词语可以标记在概率尺上。例如，概率为 0.6 的事件比不发生更可能发生，而概率为 0.1 的事件则非常不可能发生。

You should be able to mark probabilities on a number line from 0 to 1 and interpret given positions.

你应能在 0 到 1 的数轴上标出概率值，并解读给定的位置。

2. Basic Probability Formula | 基本概率公式

If all outcomes are equally likely, the probability of an event A is given by:

如果所有结果等可能发生，事件 A 的概率公式为：

P(A) = Number of favourable outcomes / Total number of outcomes

For example, when rolling a fair six-sided die, the probability of rolling an even number is 3/6 = ½, because there are three favourable outcomes (2, 4, 6) out of a total of six.

例如，投掷一个均匀的六面骰子时，掷出偶数的概率是 3/6 = ½，因为在总共六个结果中有三个有利结果（2、4、6）。

This formula is often written as 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 S.

这个公式常写作 P(A) = n(A) / n(S)，其中 n(A) 是事件 A 包含的结果数，n(S) 是样本空间 S 中的总结果数。

3. Expected Frequency | 期望频率

If you repeat an experiment a certain number of times, the expected frequency of an event is the number of times you would expect it to occur based on its theoretical probability.

如果你将某个实验重复一定次数，事件的期望频率是指根据其理论概率你预计它发生的次数。

Expected frequency = Probability × Number of trials

For instance, if you flip a fair coin 200 times, the expected number of heads is ½ × 200 = 100. This does not guarantee exactly 100 heads, but it gives a good prediction over many trials.

比如，将一枚均匀硬币抛 200 次，出现正面的期望次数是 ½ × 200 = 100。这并不保证恰好 100 次正面，但在大量试验中是一个很好的预测。

This idea is often tested alongside experimental probability to compare observed and expected results.

这个概念常与实验概率一起考查，比较观察结果与期望结果。

4. Sample Space Diagrams | 样本空间图

A sample space is a list or diagram showing all possible outcomes of an experiment. For two combined events, a sample space diagram (often a two-way table) helps you count the total outcomes and identify favourable ones.

样本空间是列出或展示一个实验所有可能结果的方式。对于两个组合事件，样本空间图（常为双向表）可帮助你数出总结果并找出有利结果。

When rolling two fair dice, the sample space can be shown as a 6×6 grid. Each cell represents one outcome, such as (2,5) for a total of 7. The total number of outcomes is 36. You can then find the probability of an event like ‘the sum is 7’ by counting the cells that satisfy the condition (there are 6) and using the basic formula.

投掷两个均匀骰子时，样本空间可用 6×6 网格表示。每个格子代表一个结果，如 (2,5) 表示和为 7。总结果数为 36。然后你可以通过数出满足条件的格子（例如和为 7 的有 6 个）并使用基本公式来计算概率。

Other useful sample space tools include lists, possibility space grids, and for more complex problems, tree diagrams or Venn diagrams.

其他有用的样本空间工具包括列表、可能性空间网格，以及对于更复杂的问题，树形图或韦恩图。

5. Mutually Exclusive Events | 互斥事件

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 on the same roll are mutually exclusive.

如果两个事件不能同时发生，则它们是互斥的。例如，掷骰子时，同一掷中既得到 2 又得到 5 就是互斥的。

For mutually exclusive events A and B, the probability of A or B occurring is the sum of their individual probabilities:

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, you must subtract the probability of them both occurring to avoid double-counting: P(A or B) = P(A) + P(B) − P(A and B). This is the general addition rule.

如果事件不是互斥的，你必须减去它们同时发生的概率以避免重复计算：P(A 或 B) = P(A) + P(B) − P(A 且 B)。这是一般的加法法则。

6. Independent Events | 独立事件

Two events are independent if the occurrence of one does not affect the probability of the other occurring. For example, flipping a coin and rolling a die are independent – the coin’s result does not change the probability of rolling a 6.

如果当一个事件的发生不影响另一个事件的发生概率，则这两个事件是独立的。例如，抛硬币和掷骰子是独立的——硬币的结果不会改变掷出 6 的概率。

For independent events A and B, the probability of both A and B occurring is the product of their probabilities:

对于独立事件 A 和 B，两者都发生的概率是各自概率的乘积：

P(A and B) = P(A) × P(B)

It is important to check carefully whether events are independent or not. In many GCSE questions, you will be told when events are independent, or you will need to deduce it from the context (e.g. ‘replacing’ a card keeps the events independent; ‘without replacement’ makes them dependent).

仔细检查事件是否独立很重要。在许多 GCSE 题目中，会告诉你事件是独立的，或者你需要从语境中推断（例如，“放回”使得事件保持独立；“不放回”则使其不独立）。

7. Tree Diagrams | 树形图

Tree diagrams are extremely useful for showing the outcomes of two or more successive events, especially when events are independent or when items are selected without replacement.

树形图在展示两个或多个连续事件的结果时极为有用，尤其是当事件独立或有放回/无放回选择时。

Each branch represents a possible outcome, labelled with its probability. To find the probability of a combination of events, you multiply along the branches. If more than one combination leads to the same final result, you add those probabilities.

每条分支代表一个可能的结果，并标注其概率。要求出某个事件组合的概率，就沿着分支相乘。如果多个组合导致相同的最终结果，就将那些概率相加。

For example, a bag contains 4 red and 6 blue counters. One counter is taken, its colour noted, and then replaced before a second counter is taken. The tree diagram will show two sets of identical branches (4/10 red, 6/10 blue). The probability of getting two reds is 4/10 × 4/10 = 16/100 = 4/25.

例如，一个袋子里有 4 个红色和 6 个蓝色筹码。取出一个记录颜色后放回，再取第二个。树形图将显示两组相同的分支（4/10 红，6/10 蓝）。得到两个红色的概率是 4/10 × 4/10 = 16/100 = 4/25。

For conditional probabilities (Higher tier only) ‘without replacement’, the probabilities on the second set of branches change depending on the first outcome. You must adjust the fractions accordingly.

对于条件概率（仅高级层级）的“无放回”情形，第二组分支上的概率会依据第一次的结果而改变。你必须相应地调整分数。

8. Conditional Probability (Higher Tier) | 条件概率（高级）

Conditional probability is the probability of an event B occurring given that event A has already occurred. It is written as P(B | A).

条件概率是指在事件 A 已经发生的情况下，事件 B 发生的概率，记作 P(B | A)。

P(B | A) = P(A and B) / P(A)

For example, a class has 12 boys and 18 girls. 8 boys and 12 girls wear glasses. If a randomly chosen student wears glasses, the probability the student is a boy is P(boy | glasses) = number of boys with glasses / total number of students with glasses = 8/20 = 2/5.

例如，一个班级有 12 个男生和 18 个女生。其中 8 个男生和 12 个女生戴眼镜。若随机选出一名学生且该生戴眼镜，这名学生是男生的概率为 P(男生 | 眼镜) = 戴眼镜的男生数 / 戴眼镜的总人数 = 8/20 = 2/5。

Tree diagrams are often the clearest way to handle conditional probability questions, because you can directly see how the probabilities change after a selection without replacement.

树形图通常是处理条件概率问题最清晰的方式，因为你可以直接看到无放回选择后概率如何变化。

9. Venn Diagrams and Set Notation | 韦恩图与集合表示

Venn diagrams use overlapping circles to show the relationships between sets. In probability, the universal set ε contains all possible outcomes. Each event is a subset of ε. You might be asked to shade regions such as A ∩ B (intersection), A ∪ B (union), or A’ (complement).

韦恩图用重叠的圆圈表示集合之间的关系。在概率中，全集 ε 包含所有可能结果，每个事件是 ε 的子集。你可能会被要求给某些区域涂色，比如 A ∩ B（交集）、A ∪ B（并集）或 A’（补集）。

The probability of an event can be found from a Venn diagram by adding up the numbers in the relevant region(s) and dividing by the total number of items in the universal set. For example, P(A ∪ B) = (elements in A or B or both) / total elements.

通过将相关区域内的数相加并除以全集中元素的总数，可以从韦恩图中找出事件的概率。例如，P(A ∪ B) =（在 A 或 B 或两者中的元素数）/ 总元素数。

Common set notation includes: ε universal set, ∅ empty set, ∈ ‘is an element of’, ∪ union, ∩ intersection, ‘ complement. GCSE Higher tier candidates should be confident reading and using this notation.

常见的集合表示法包括：ε 全集，∅ 空集，∈ “属于”，∪ 并集，∩ 交集，’ 补集。GCSE 高级层级的考生应能熟练阅读和使用这些符号。

10. Two-Way Tables and Frequency Trees | 双向表与频率树

Two-way tables (also called contingency tables) are a tidy way to display frequencies for two categorical variables. They allow you to find probabilities, including conditional probabilities, by reading off the appropriate row and column totals.

双向表（也叫列联表）是展示两个分类变量频率的整洁方式。通过读出相应的行和列总数，你可以从中找出概率，包括条件概率。

For example, a table showing gender and left/right-handedness lets you quickly compute the probability a randomly chosen person is left-handed, or the probability a left-handed person is female.

例如，一个显示性别与左/右撇子的表格能让你快速计算随机一人是左撇子的概率，或一个左撇子是女性的概率。

Frequency trees work similarly to tree diagrams but show actual frequencies rather than probabilities. They are especially useful when all data is given as counts, and you need to deduce missing values before finding probabilities.

频率树的工作原理与树形图类似，但显示的是实际频数而非概率。当所有数据以计数给出，且你需要先推导缺失值再求概率时，频率树特别有用。

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

When you cannot determine a theoretical probability, you can estimate it using an experiment or historical data. The relative frequency of an event is:

当你无法确定理论概率时，可以通过实验或历史数据来估计。事件的相对频率为：

Relative frequency = Number of times event occurs / Total number of trials

As the number of trials increases, the relative frequency tends to get closer to the theoretical probability, if one exists. This is often called the law of large numbers.

随着试验次数增多，相对频率会趋向于接近理论概率（如果存在理论概率的话）。这通常被称为大数定律。

GCSE questions might ask you to compare an observed relative frequency with a claimed theoretical probability and discuss whether the experimental data supports the claim. You must comment on the sample size – larger samples give more reliable estimates.

GCSE 题目可能会要求你比较观察到的相对频率与声称的理论概率，并讨论实验数据是否支持该说法。你必须对样本量进行评论——更大的样本量会提供更可靠的估计。

12. Exam Tips for Probability | 概率考试技巧

Always check whether probabilities sum to 1. In a completed tree diagram, the probabilities on branches from the same point should add up to 1. On a Venn diagram, all probabilities in the diagram should total 1.

务必检查概率之和是否为 1。在完成的树形图中，同一点分出的分支概率之和应等于 1。在韦恩图中，图中所有概率总和应为 1。

Simplify fractions where possible, but often it is acceptable to leave them unsimplified if the question does not insist. However, a final answer is best given in its simplest form.

可能的话要化简分数，但如果题目不坚持，不化简通常也可接受。不过，最终答案最好以最简形式给出。

When working with tree diagrams, clearly label the end-of-branch outcomes with both the combination and the probability. Show your multiplications and additions step by step to earn method marks even if you make an arithmetic error.

在使用树形图时，清楚地用组合和概率标出分支末端的结果。一步一步展示你的乘法和加法，这样即使计算错误，也能获得方法分。

For ‘without replacement’ problems, write out the new totals at each stage. If a bag has 10 counters and one red is removed, the next draw has a total of 9 counters, not 10. Many marks are lost by forgetting to update the denominator.

对于“无放回”问题，要在每个阶段写出新的总数。如果一个袋子里有 10 个筹码，取走一个红色，下一次抽取的总数是 9 而不是 10。很多分数都是因为忘记更新分母而失去的。

Finally, interpret your result in the context of the question. If asked whether a game is fair, calculate the expected gain/loss or compare probabilities. A fair game gives an expected outcome of zero.

最后，要结合题目背景解释你的结果。如果被问及游戏是否公平，计算期望收益/损失或比较概率。公平的游戏的期望结果应为零。

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GCSE Maths: Probability Revision Guide | GCSE 数学：概率 考点精讲