📚 GCSE AQA Maths: Probability Revision | GCSE AQA 数学：概率考点精讲
Probability is the mathematical study of chance and uncertainty. It measures how likely an event is to happen, using a scale from 0 (impossible) to 1 (certain). In GCSE AQA Mathematics, you will learn to express probability as a fraction, decimal or percentage, and to use diagrams, tables and rules to solve multi-event problems. Mastering these concepts is essential for both foundation and higher tier papers, as probability questions appear regularly and often combine several topics.
概率是对机会和不确定性的数学研究。它衡量一个事件发生的可能性，使用从 0（不可能）到 1（必然）的尺度。在 GCSE AQA 数学中，你将学习用分数、小数或百分数表达概率，并利用图表、表格和法则解决多事件问题。掌握这些概念对基础卷和进阶卷都至关重要，因为概率题目经常出现，且常综合多个主题。
1. Basic Probability | 基本概率
The probability of an event E is given by: Number of favourable outcomes divided by the total number of possible outcomes, provided all outcomes are equally likely. This can be written as P(E) = n(E)/n(S), where S is the sample space. Probabilities always lie between 0 and 1 inclusive. A probability of 0 means the event is impossible; a probability of 1 means it is certain. For example, when rolling a fair six-sided die, the probability of rolling a 4 is 1/6.
事件 E 的概率等于：有利结果的数量除以所有可能结果的总数，前提是所有结果等可能。可写作 P(E) = n(E)/n(S)，其中 S 为样本空间。概率值始终在 0 到 1 之间（含端点）。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。例如，掷一枚均匀的六面骰子，掷出 4 的概率是 1/6。
Probabilities can be written as fractions, decimals or percentages. The sum of probabilities of all mutually exclusive outcomes of an experiment equals 1. So if the probability of an event happening is p, the probability of it not happening is 1 – p. This is called the complement rule.
概率可以用分数、小数或百分数表示。一个实验中所有互斥结果的概率之和等于 1。因此，若某事件发生的概率为 p，则不发生的概率为 1 – p。这称为互补规则。
2. Sample Space Diagrams | 样本空间图
A sample space diagram lists all possible outcomes of an experiment systematically. For two events, such as flipping a coin and rolling a die, you can draw a grid: coin outcomes (H, T) as rows and die outcomes (1–6) as columns, giving 12 equally likely pairs. Each cell represents one outcome, so P(heads and a 3) = 1/12. Sample space grids help ensure no outcome is missed when calculating probabilities.
样本空间图系统地列出实验的所有可能结果。对于两个事件，例如抛一枚硬币和掷一个骰子，可绘制网格：硬币结果（H、T）为行，骰子结果（1–6）为列，得到 12 个等可能的配对。每个格子代表一个结果，故 P(正面且 3) = 1/12。样本空间网格有助于在计算概率时不遗漏任何结果。
You can also use a sample space for combined events like two spinners or two dice. For two dice, a 6×6 grid shows 36 outcomes. The probability of scoring a sum of 7 is 6/36 = 1/6, because the pairs (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) all give a sum of 7. Such grids are especially useful for finding probabilities of ‘and’ events when the events are independent.
你还可以为复合事件使用样本空间，比如两个转盘或两粒骰子。对于两粒骰子，6×6 的网格显示 36 种结果。总点数为 7 的概率是 6/36 = 1/6，因为 (1,6)、(2,5)、(3,4)、(4,3)、(5,2)、(6,1) 都给出和为 7。当事件独立时，这种网格对求“且”事件的概率特别有用。
3. Expected Frequency | 期望频数
Expected frequency predicts how many times an event is likely to occur in a certain number of trials. It is calculated by multiplying the probability of the event by the number of trials:
Expected frequency = Probability × Number of trials
期望频数预测某事件在特定试验次数中大概会发生多少次。计算方法为：事件概率乘以试验次数：
期望频数 = 概率 × 试验次数
If you toss a fair coin 200 times, the expected number of heads is 0.5 × 200 = 100. Note that this is only an average prediction; actual results may vary. Expected frequencies are widely used in probability estimates and in comparing experimental data with theoretical models.
如果抛一枚均匀硬币 200 次，正面朝上的期望次数是 0.5 × 200 = 100。注意这只是平均预测，实际结果可能有所不同。期望频数广泛应用于概率估算以及实验数据与理论模型的比较。
4. Relative Frequency | 相对频率
Relative frequency is an estimate of probability based on experimental data. It is calculated using the formula:
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. This is known as the law of large numbers. For instance, if a biased coin is tossed 300 times and lands on heads 165 times, the relative frequency of heads is 165/300 = 0.55, suggesting the probability of heads is about 0.55.
相对频率是基于实验数据对概率的估计。计算公式为：
相对频率 = 事件发生次数 ÷ 总试验次数
随着试验次数增加，相对频率倾向于接近理论概率，这就是大数定律。例如，一枚偏斜的硬币被抛掷 300 次，正面朝上 165 次，则正面的相对频率为 165/300 = 0.55，暗示正面的概率大约为 0.55。
You are expected to use relative frequency to estimate probabilities and to evaluate whether a dice or coin is fair. If the relative frequency after many trials is far from the theoretical probability, the object may be biased.
你应能利用相对频率估算概率，并判断骰子或硬币是否均匀。若大量试验后的相对频率与理论概率相差甚远，则该物品可能是偏斜的。
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, the events ‘rolling a 2’ and ‘rolling an odd number’ are mutually exclusive, but ‘rolling an even number’ and ‘rolling a number greater than 4’ are not, because 6 satisfies both.
如果两个事件不可能同时发生，则它们是互斥的。例如，掷骰子时，“掷出 2”和“掷出奇数”是互斥的，但“掷出偶数”和“掷出大于 4 的数”不互斥，因为 6 同时满足两个条件。
For mutually exclusive events A and B, the probability that A or B occurs is the sum of their individual probabilities:
P(A or B) = P(A) + P(B)
This is the addition rule for mutually exclusive events. If events are not mutually exclusive, you must subtract the overlap to avoid double counting.
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 或 B) = P(A) + P(B)
这是互斥事件的加法法则。若事件不互斥，则须减去重叠部分以避免重复计算。
6. Independent Events & Tree Diagrams | 独立事件与树状图
Independent events are those where the outcome of one event does not affect the outcome of another. For example, flipping a coin and rolling a die are independent. For independent events A and B, the probability of both occurring is:
P(A and B) = P(A) × P(B)
独立事件是指一个事件的结果不影响另一个事件结果的事件。例如，抛硬币和掷骰子相互独立。对于独立事件 A 和 B，两者同时发生的概率为：
P(A 且 B) = P(A) × P(B)
Tree diagrams are a powerful tool for showing all possible outcomes of multiple independent events. Each branch represents an outcome and is labelled with its probability. To find the probability of a specific combination of events, multiply the probabilities along the path. For example, a bag contains 3 red and 2 blue marbles. If a marble is drawn, replaced, and drawn again, the probability of drawing red then blue is 3/5 × 2/5 = 6/25. Always check that the probabilities on branches from the same point sum to 1.
树状图是展示多个独立事件所有可能结果的有力工具。每个分支代表一个结果并标注其概率。要找到特定组合事件的概率，将路径上的概率相乘。例如，袋中有 3 颗红弹珠和 2 颗蓝弹珠。如果抽取一颗，放回后再抽一颗，那么先红后蓝的概率是 3/5 × 2/5 = 6/25。始终检查同一点出发的分支概率之和是否为 1。
7. Conditional Probability | 条件概率
Conditional probability is the probability of an event occurring given that another event has already happened. It is written P(B|A), meaning “the probability of B given A”. For dependent events, the outcome of the first event affects the probability of the second. The multiplication rule changes to: P(A and B) = P(A) × P(B|A).
条件概率是指在另一事件已发生的情况下，某事件发生的概率。写作 P(B|A)，意为“在 A 发生的条件下 B 的概率”。对于相关事件，第一个事件的结果会影响第二个事件的概率。乘法法则变为：P(A 且 B) = P(A) × P(B|A)。
Tree diagrams for conditional probability show different probabilities on the second set of branches depending on the outcome of the first event. For example, picking two sweets from a bag without replacement: if the bag contains 4 toffees and 6 chocolates, P(first choc) = 6/10. If a chocolate is taken, 5 chocolates remain out of 9 sweets, so P(second choc | first choc) = 5/9. The probability of two chocolates is 6/10 × 5/9 = 30/90 = 1/3.
条件概率的树状图会根据第一个事件的结果在第二组分枝上显示不同的概率。例如，不放回地从袋中取两颗糖：若袋中有 4 颗太妃糖和 6 颗巧克力糖，P(第一颗是巧克力) = 6/10。若第一颗取出巧克力，则剩下 9 颗糖中有 5 颗巧克力，故 P(第二颗是巧克力 | 第一颗是巧克力) = 5/9。两颗皆为巧克力的概率为 6/10 × 5/9 = 30/90 = 1/3。
You can also use the formula P(B|A) = P(A and B) / P(A) when you have combined probabilities from a table or Venn diagram. This is essential for higher tier exam questions.
当从表格或维恩图中获得联合概率时，也可使用公式 P(B|A) = P(A 且 B) / P(A)。这对进阶卷的考试题至关重要。
8. Venn Diagrams & Probability | 维恩图与概率
A Venn diagram shows sets and their overlaps, which is useful for events that are not mutually exclusive. Each rectangle represents the sample space, and circles represent events. The overlap region shows outcomes in both A and B (A ∩ B). The numbers in each region are the number of outcomes or probabilities. From a completed Venn diagram, you can find probabilities like P(A), P(B), P(A ∩ B) and P(A ∪ B).
维恩图展示集合及其重叠部分，对非互斥的事件十分有用。矩形代表样本空间，圆代表事件。重叠区域表示既在 A 又在 B 中的结果（A ∩ B）。每个区域内的数字是结果数或概率。从填充完整的维恩图中，可以求得 P(A)、P(B)、P(A ∩ B) 和 P(A ∪ B) 等概率。
The probability of A union B is P(A ∪ B) = P(A) + P(B) – P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0 so this simplifies to P(A) + P(B). You must be able to complete a Venn diagram given set data and then answer probability questions based on it.
A 与 B 的并集概率为 P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。对于互斥事件，P(A ∩ B) = 0，因此简化为 P(A) + P(B)。你必须能根据给定的集合数据补全维恩图，然后基于它回答概率问题。
9. The AND/OR Rules | 乘法与加法法则
These rules summarise how to combine probabilities for compound events. For OR: if events are mutually exclusive, use P(A or B) = P(A) + P(B). If not mutually exclusive, use P(A or B) = P(A) + P(B) – P(A and B). For AND: if events are independent, use P(A and B) = P(A) × P(B). If dependent, use P(A and B) = P(A) × P(B|A). Always check the conditions before applying a rule.
这些法则总结了如何组合复合事件的概率。对于“或”：若事件互斥，则用 P(A 或 B) = P(A) + P(B)；若不互斥，则用 P(A 或 B) = P(A) + P(B) – P(A 且 B)。对于“且”：若事件独立，则用 P(A 且 B) = P(A) × P(B)；若相关，则用 P(A 且 B) = P(A) × P(B|A)。应用法则前务必检查条件。
A common exam question gives a scenario and asks for probabilities like “the probability that it rains on both Saturday and Sunday” or “the probability that a student studies at least one language”. Breaking the problem down using these rules makes it manageable.
常见考题会给出一个情境，要求计算如“周六和周日都下雨的概率”或“某学生至少学习一门语言的概率”。利用这些法则分解问题会使解答变得可控。
10. Using Two-Way Tables | 使用双向表
Two-way tables organise data about two categorical variables and are ideal for finding conditional probabilities and combined events. For instance, a table might show students who play football and basketball. The totals of each row and column give the marginal probabilities, and a cell gives the joint probability. From such a table, you can find P(plays football and basketball), P(plays football | plays basketball), or P(plays neither).
双向表将两个分类变量的数据组织起来，特别适合求条件概率和组合事件。例如，一张表可以显示参加足球和篮球的学生人数。每行每列的总和给出边缘概率，而单元格给出联合概率。从这样的表中，你可以求得 P(踢足球且打篮球)、P(踢足球 | 打篮球) 或 P(两者都不参加)。
To find a conditional probability from a two-way table, restrict attention to the row or column of the given condition and calculate the fraction within that group. This method is often tested at higher tier and is a reliable way to avoid mistakes with complex wording.
要从双向表中求条件概率，只需将注意力限制在给定条件的行或列，并计算该组内的比例。这种方法常在进阶卷中考查，是避免复杂文字题出错的有效方式。
11. Probability with Algebra | 代数概率
Sometimes probability problems involve unknown quantities that you must find using algebra. For instance, a bag contains x red balls, y blue balls and z green balls. If you are given probabilities like P(red) = 2/5 and total number of balls is 40, you can set up equations to solve for x, y, z. Setting up P(red) = x/(x+y+z) = 2/5 and using total = 40 gives x = 16, and so on.
有时概率问题会涉及未知量，需要用代数求解。例如，袋中有 x 个红球、y 个蓝球和 z 个绿球。如果给出的概率如 P(红) = 2/5，且球总数是 40，你可以建立方程求解 x、y、z。设 P(红) = x/(x+y+z) = 2/5，结合总数为 40 可得 x = 16，以此类推。
More challenging questions involve tree diagrams with algebraic probabilities on branches (e.g., w and 1–w) and then asking to find w by using the fact that the probabilities of all final outcomes sum to 1. This tests your ability to set up and solve linear or quadratic equations from probability scenarios.
更具挑战性的题目会在树状图的分支上使用代数概率（如 w 和 1–w），然后要求利用所有最终结果的概率之和为 1 来求 w。这考验你根据概率情境建立并求解一次或二次方程的能力。
12. Exam Tips | 考试技巧
Always express your final probability in its simplest form unless the question specifies otherwise. A fraction like 4/8 should be simplified to ½. If a question asks for ‘chance’, ‘likelihood’, or ‘risk’, these all refer to probability and can be answered in fraction, decimal or percentage form. Read the question carefully to see whether items are replaced or not, as this affects independence and tree diagram probabilities.
除非题目另有要求，始终将最终概率化为最简形式。分数如 4/8 应化简为 ½。如果题目提到“机会”、“可能性”或“风险”，这些都指概率，可用分数、小数或百分数回答。仔细读题，看清物品是否有放回，因为这会影响到独立性和树状图的概率。
For foundation tier, focus on basic probability, sample space, relative frequency and simple tree diagrams. For higher tier, practise conditional probability, Venn diagrams with algebraic unknowns, and interpreting two-way tables. Show all working, especially the multiplication along tree branches, as marks are awarded for method. Finally, check that probabilities are between 0 and 1 and that the sum of mutually exclusive outcomes equals 1.
基础卷的重点是基本概率、样本空间、相对频率和简单树状图。进阶卷则要练习条件概率、含代数未知量的维恩图以及双向表的解读。展示所有步骤，尤其是树状图分支上的乘法，因为方法也能得分。最后检查概率是否在 0 到 1 之间，且互斥结果的概率之和是否为 1。
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