GCSE Edexcel Maths: Probability Essentials | GCSE Edexcel 数学：概率考点精讲

📚 GCSE Edexcel Maths: Probability Essentials | GCSE Edexcel 数学：概率考点精讲

Probability is the branch of mathematics that measures the chance of an event happening. For GCSE Edexcel Maths, you need to understand basic probability principles, sample spaces, tree diagrams, and conditional probability, as well as using set notation and frequency tables to solve problems. This guide covers all key topics with clear explanations.

概率是数学中衡量事件发生可能性的分支。在 GCSE Edexcel 数学考试中，你需要理解基本概率原理、样本空间、树状图、条件概率，并能用集合符号和频率表解决问题。本指南涵盖所有关键主题并附有清晰解释。

1. Basic Probability Concepts | 基础概率概念

The probability of an event is a number between 0 and 1, where 0 means impossible and 1 means certain. It can be written as a fraction, decimal, or percentage. For equally likely outcomes, probability = (number of favourable outcomes) / (total number of outcomes).

事件的概率是介于 0 和 1 之间的一个数字，0 表示不可能，1 表示必然发生。概率可以用分数、小数或百分比表示。对于等可能结果，概率 = (有利结果的数量) / (总结果数量)。

The probability of an event A is denoted by P(A). The sum of probabilities of all mutually exclusive outcomes in a sample space is 1. So if A is an event, the probability of A not happening is P(not A) = 1 – P(A).

事件 A 的概率记作 P(A)。样本空间中所有互斥结果的概率和为 1。因此如果 A 是一个事件，那么 A 不发生的概率是 P(非 A) = 1 – P(A)。

Probabilities can be compared on a probability scale, marked from 0 to 1, with words like impossible, unlikely, even chance, likely, certain.

可以在概率标尺上比较概率，标尺从 0 到 1，用词语如不可能、不太可能、对等机会、很可能、必然。

2. Sample Spaces and Listing Outcomes | 样本空间与列出结果

A sample space is the set of all possible outcomes of an experiment. You can list outcomes systematically using a table, list, or grid. For two dice, a sample space diagram shows all 36 possible combinations.

样本空间是一个实验所有可能结果的集合。你可以用表格、列表或网格系统地列出结果。对于两个骰子，样本空间图显示所有 36 种可能的组合。

Example: When flipping two coins, the sample space is {HH, HT, TH, TT}. Each outcome has probability ¼ if the coins are fair.

例子：抛两枚硬币时，样本空间是 {HH, HT, TH, TT}。如果硬币公平，每种结果的概率是 1/4。

For combined events, a two-way table or a list helps find probabilities such as the sum of two dice being 7. (6 outcomes out of 36, so probability = 6/36 = 1/6).

对于组合事件，双向表或列表有助于求出概率，比如两个骰子点数之和为 7（36 种中有 6 种，所以概率 = 6/36 = 1/6）。

3. Mutually Exclusive and Independent Events | 互斥事件与独立事件

Mutually exclusive events cannot occur at the same time. If A and B are mutually exclusive, then P(A or B) = P(A) + P(B). This is the addition rule for mutually exclusive events. For example, rolling a 3 and rolling a 5 on a dice are mutually exclusive.

互斥事件不能同时发生。如果 A 和 B 互斥，则 P(A 或 B) = P(A) + P(B)。这是互斥事件的加法法则。例如，掷骰子得到 3 和得到 5 是互斥的。

When events are not mutually exclusive, we must subtract the intersection: P(A or B) = P(A) + P(B) – P(A and B). Use this when there is an overlap, e.g. drawing a red card or a king from a pack of cards.

当事件不互斥时，我们必须减去重叠部分：P(A 或 B) = P(A) + P(B) – P(A 和 B)。有重叠时使用，例如从一副牌中抽出一张红色牌或一张 K。

Independent events are those where one does not affect the probability of the other. For independent events, P(A and B) = P(A) × P(B). Typical examples: flipping a coin and rolling a dice.

独立事件是一个事件的发生不影响另一个事件概率的事件。对于独立事件，P(A 和 B) = P(A) × P(B)。常见例子：抛硬币和掷骰子。

In contrast, if events are not independent, you need to consider conditional probability, which we will discuss later.

相反，如果事件不是独立的，则需要考虑条件概率，稍后讨论。

4. Venn Diagrams and Set Notation | 维恩图与集合符号

Venn diagrams show events as circles inside a rectangle representing the sample space (universal set ξ). Overlap regions represent intersections (A ∩ B), the whole of both circles is the union (A ∪ B). The outside area is the complement (Aᶜ or not A).

维恩图将事件显示为长方形（代表样本空间，全集合 ξ）内的圆。重叠区域表示交集 (A ∩ B)，两个圆整体表示并集 (A ∪ B)。外部区域表示补集 (Aᶜ 或非 A)。

Set notation is essential in Edexcel exam questions: n(A) means number of elements in set A. P(A ∪ B) = P(A) + P(B) – P(A ∩ B). You may be asked to shade regions or find probabilities from given numbers in each region.

集合符号在 Edexcel 考试题中至关重要：n(A) 表示集合 A 的元素数量。P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。可能会要求你给区域涂色，或根据图中各区域的数字求概率。

For three sets, Venn diagrams become more complex, including intersections like A ∩ B ∩ C. Always fill from the centre intersection outward when placing numbers.

对于三个集合，维恩图变得更为复杂，包括 A ∩ B ∩ C 等交集。放置数字时始终从中心交集向外填写。

5. Two-way Tables and Frequency Trees | 双向表与频率树

Two-way tables summarise data for two categorical variables. They help find probabilities of combined events, including conditional probabilities. For example, a table of 100 students by gender and whether they play an instrument.

双向表汇总两个分类变量的数据。它们有助于求出组合事件的概率，包括条件概率。例如，一张 100 名学生的表格，按性别和是否学习乐器分类。

To find the probability of a student playing an instrument given they are female, you restrict to the ‘female’ row and use the count in the ‘plays instrument’ column divided by the total in that row.

要求出已知是女生时学习乐器的概率，你限制在“女生”行，用“学习乐器”列的人数除以该行的总数。

Frequency trees record the number of items at each stage of a sequence of events, without probabilities. They help calculate frequencies before converting to probabilities. For instance, a frequency tree for 200 people, categorised by whether they own a car and a bicycle.

频率树记录事件序列每个阶段的项目数量，不含概率。它们有助于先计算频率，再转换为概率。例如，对 200 人按是否拥有汽车和自行车进行分类的频率树。

In the exam, you may need to complete a frequency tree and then use the numbers to find a probability — a straightforward method that avoids complex formulas.

考试中可能需要你完成频率树，然后用数字求出概率——一种避免复杂公式的简单方法。

6. Tree Diagrams | 树状图

Tree diagrams show sequences of two or more events. Each branch is labelled with its probability. Probabilities of combined outcomes are found by multiplying along the branches. The sum of probabilities from a node is 1.

树状图显示两个或多个事件的序列。每个分枝标有其概率。组合结果的概率通过沿分枝相乘求得。从一个节点出发的所有概率之和为 1。

Tree diagrams are used for both independent events (like tossing a coin and rolling a die) and dependent events (like picking counters without replacement).

树状图适用于独立事件（如抛硬币和掷骰子）和非独立事件（如不放回抽选筹码）。

To find the probability of at least one success, it is often easier to calculate 1 – P(all failures). Multiply along the failure branches and subtract from 1.

要求出至少一次成功的概率，通常更容易计算 1 – P(全部失败)。沿失败分枝相乘并从 1 中减去。

Example: A bag has 3 red and 5 blue counters. Two counters are taken without replacement. Draw a tree with first pick (3/8 red, 5/8 blue) and second pick probabilities depending on the first. The probability of picking one of each colour is found by adding the two mixed pathway products.

例子：一个袋子有 3 个红色和 5 个蓝色筹码。不放回地取两次。绘制一个树状图，第一次抽取（3/8 红，5/8 蓝），第二次概率根据第一次而定。抽得一红一蓝的概率通过将两条混合路径的乘积相加得到。

7. Conditional Probability | 条件概率

Conditional probability is the probability of an event B occurring given that event A has already occurred, written P(B|A). It is defined by the formula:

条件概率是在事件 A 已经发生的条件下事件 B 发生的概率，记为 P(B|A)。其公式定义为：

P(B|A) = P(A ∩ B) / P(A)

This formula is provided on the Edexcel formula sheet. You can use it directly or deduce probabilities from a Venn diagram, two-way table, or tree diagram.

这个公式在 Edexcel 公式表中提供。你可以直接使用，或从维恩图、双向表或树状图中推导概率。

On tree diagrams, the second set of branches shows conditional probabilities. For example, if the first counter is red, the probability of the second being blue is P(blue | red was first).

在树状图中，第二组分枝显示条件概率。例如，如果第一个筹码是红色，第二个是蓝色的概率是 P(蓝 | 第一个是红)。

Always read the question carefully: ‘given that’ signals a conditional probability. The sample space is reduced to only those outcomes where the condition is satisfied.

始终仔细读题：“已知”表示条件概率。样本空间缩减到仅满足条件的那些结果。

Example: A class has 12 boys and 8 girls. 5 boys wear glasses and 3 girls wear glasses. Find P(boy | wears glasses). Restrict the sample space to glasses-wearers (5+3=8), then P = 5/8.

例子：班级有 12 名男孩和 8 名女孩。5 名男孩戴眼镜，3 名女孩戴眼镜。求 P(男孩|戴眼镜)。将样本空间限制到戴眼镜者 (5+3=8)，则 P = 5/8。

8. Relative Frequency and Expected Frequency | 相对频率与期望频率

Relative frequency is an estimate of probability based on experiments or historical data: relative frequency = (number of successful trials) / (total number of trials). The more trials, the closer the relative frequency tends to the theoretical probability.

相对频率是基于实验或历史数据对概率的估计：相对频率 = (成功试验次数) / (总试验次数)。试验次数越多，相对频率越接近理论概率。

Expected frequency is the number of times an event is expected to happen in a number of trials: expected frequency = probability × number of trials. For example, if the probability of rain is 0.3, in 50 days we expect rain on 0.3×50 = 15 days.

期望频率是在若干次试验中预期事件发生的次数：期望频率 = 概率 × 试验次数。例如，如果降雨概率是 0.3，在 50 天中我们预期降雨 0.3×50 = 15 天。

In Edexcel questions, you might be given a frequency table of experimental results and asked to compare relative frequency with theoretical probability, or to test a claim about a biased dice.

在 Edexcel 考题中，可能会给你实验结果频率表，要求比较相对频率与理论概率，或检验关于一个有偏骰子的说法。

Remember: a relative frequency diagram can be used to see whether the experimental probability stabilises around a certain value after many trials.

记住：相对频率图可用于观察多次试验后实验概率是否稳定在某个值附近。

9. Probability and Counting – Product Rule | 概率与计数——乘积法则

For independent combined events, the total number of outcomes is the product of the number of outcomes of each event (the product rule for counting). For example, a meal deal with 3 sandwich choices, 4 drink choices, and 2 dessert choices has 3 × 4 × 2 = 24 possible combinations.

对于独立的组合事件，总结果数是每个事件结果数的乘积（计数乘积法则）。例如，一个套餐有 3 种三明治、4 种饮料和 2 种甜点，共有 3 × 4 × 2 = 24 种可能的组合。

This principle links directly to probability: if all outcomes are equally likely, the probability of a specific combination is 1 / (total number of outcomes).

这一原则与概率直接相关：如果所有结果等可能，则特定组合的概率是 1 / (总结果数)。

In more complex scenarios, you can list outcomes systematically (like using a sample space diagram) or apply the product rule to find the total sample space size before calculating probabilities.

在更复杂的情况下，你可以系统列出结果（如使用样本空间图）或应用乘积法则求出样本空间总大小，再计算概率。

10. Algebra in Probability Problems | 概率问题中的代数

Sometimes probabilities are given as algebraic expressions, for example P(A) = x, P(not A) = 3x + 0.2. Since these sum to 1, we form an equation x + (3x + 0.2) = 1, solve for x, then find the required probability.

有时概率以代数表达式给出，例如 P(A) = x，P(非 A) = 3x + 0.2。由于它们的和为 1，我们构成方程 x + (3x + 0.2) = 1，解出 x，然后求出所需概率。

Another common type: probabilities of outcomes from a biased spinner are given in terms of k, like k, 2k, 3k, 4k. Since they sum to 1, k + 2k + 3k + 4k = 1 → 10k = 1 → k = 0.1.

另一种常见类型：一个有偏转盘的结果概率用 k 表示，如 k, 2k, 3k, 4k。由于它们和为 1，k + 2k + 3k + 4k = 1 → 10k = 1 → k = 0.1。

You may also need to set up equations from tree diagram end probabilities or frequency tables where the total number of items is unknown. Use the fact that probabilities sum to 1 or frequencies sum to the total.

你可能还需要根据树状图终点概率或总项数未知的频率表建立方程。利用概率和为 1 或频率和为总数这一事实。

Practise solving these algebraic setups quickly and accurately; they are common on higher-tier papers.

请练习快速准确地解决此类代数设置；它们常见于高等级试卷。

11. Interpreting Probability in Context | 在语境中解读概率

GCSE questions often ask whether a given probability is ‘fair’ or whether a game is ‘biased’. For a fair game, the expected winnings should be zero (or equal for all players). Compare theoretical probability to observed relative frequency to determine if a die is biased.

GCSE 题目常问某个概率是否“公平”或游戏是否“有偏”。对于公平游戏，期望赢利应为零（或所有玩家均等）。比较理论概率与观察到的相对频率以判断骰子是否有偏。

When a statement says ‘the probability of getting a 6 on this dice is 0.25’, you might need to test this with experimental data and comment on reliability. More trials give a more reliable estimate.

当题目说“这个骰子掷出 6 的概率是 0.25”时，你可能需要用实验数据检验并评论可靠性。更多试验给出更可靠的估计。

You should also be able to criticise statements such as “I haven’t rolled a six for a while, so one is due.” Each roll is independent; past outcomes do not affect future probability.

你还应能够批判如下说法：“我好久没掷出六了，所以该来了。”每次掷骰独立；过去的结果不影响未来概率。

Strong contextual understanding will secure those final marks in AO3 (interpretation and evaluation) questions.

牢固的语境理解将确保你在 AO3（解读与评价）题目中获得最后那些分数。

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

Always check whether events are independent or not. For ‘without replacement’, probabilities change for the second pick — use a tree diagram with updated branch probabilities.

始终检查事件是否独立。对于“不放回”，第二次抽取的概率改变——使用更新分支概率的树状图。

Write probabilities as fractions in simplest form unless the question asks for a decimal or percentage. Do not leave them as unsimplified fractions unless accepted.

用最简分数写概率，除非题目要求小数或百分比。不要写成未化简的分数，除非题目接受。

For conditional probability from a table, highlight the row or column that represents the condition. This reduces the risk of using the wrong total.

对于从表格中求条件概率，高亮代表条件的行或列。这有助于减少使用错误总数的风险。

In Venn diagram questions, remember: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Do not double count the intersection.

在维恩图题目中，记住：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。不要重复计算交集。

If asked to show that two events are independent, check if P(A) × P(B) = P(A ∩ B). Or verify P(B|A) = P(B). Both methods are acceptable.

如果要求证明两个事件独立，检查 P(A) × P(B) = P(A ∩ B) 是否成立。或者验证 P(B|A) = P(B) 是否成立。两种方法均可。

Finally, manage time wisely: probability questions often carry several marks but are methodical. Set out your work clearly to gain marks even if arithmetic slips.

最后，合理分配时间：概率题目通常占较多分但有条不紊。清晰列出解题步骤，即使计算失误也能得分。

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GCSE Edexcel Maths: Probability Essentials | GCSE Edexcel 数学：概率 考点精讲