📚 GCSE CIE Maths Probability Revision | GCSE CIE 数学：概率 考点精讲
Probability is a core topic in the CIE IGCSE Mathematics syllabus (0580/0980), demanding clear understanding of random experiments, sample spaces, and the rules governing chance. This article consolidates all essential concepts, from basic notation to conditional probability and tree diagrams, providing bilingual explanations and worked-style key points to help you excel in both the Core and Extended tiers.
概率是 CIE IGCSE 数学（0580/0980）的核心主题，要求清晰理解随机实验、样本空间和支配偶然性的规则。本文整合了所有基本概念，从基本符号到条件概率和树形图，提供双语解释和工作原理式的要点，助你在核心和拓展层级的考试中脱颖而出。
1. Basic Probability and the Probability Scale | 基础概率与概率标度
Probability measures how likely an event is to happen. It is always a number between 0 and 1 inclusive. A probability of 0 means impossible; 1 means certain. On the probability scale, you place events according to their likelihood, often marking probabilities on a line from 0 to 1.
概率衡量事件发生的可能性大小，总是介于 0 和 1 之间的数字（含端点）。0 表示不可能；1 表示必然发生。在概率标度上，你根据可能性放置事件，常会在一根从 0 到 1 的线上标记概率。
e outcomes, provided all outcomes are equally likely. For example, when rolling a fair six‑sided die, P(rolling a 4) = ⅙.
事件 E 的概率计算公式：P(E) = 有利结果数 / 所有可能结果总数，前提是所有结果等可能。例如，掷一枚公平的六面骰子，P(掷出 4) = ⅙。
Complementary events: P(not E) = 1 – seful shortcuts. If the probability of raining tomorrow is 0.3, then P(not raining) = 0.7.
互补事件：P(非 E) = 1 – P(E)。这是最有用的捷径之一。如果明天下雨的概率是 0.3，那么 P(不下雨) = 0.7。
2. Sample Space and Enumeration | 样本空间与列举
possible outcomes of an experiment. You may need to list outcomes systematically. For two‑ events. Tossing two coins: possible outcomes are HH, HT, TH, TT. P(at least one head) = ¾.
样本空间是实验所有可能结果的集合。你可能需要系统地列出结果。对于双向活动，样本空间图（网格）有助于找出组合事件的概率。抛两枚硬币：可能结果为 HH, HT, TH, TT。P(至少一次正面) = ¾。
When listing outcomes, avoid miss‑counts. Use a table or a list with a logical order. For a spinner numbered 1‑ comes. Each cell represents one combined outcome, all equally likely if the spinner and coin are fair.
列举结果时要避免计数错误。使用表格或按逻辑顺序列出。对于一个标有 1‑4 的转盘和一枚硬币，网格大小为 4 × 2 = 8 个结果。若转盘和硬币公平，每个单元格代表一个等可能的组合结果。
3. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法法则
ive events. Selecting a red card or a black card from a standard deck: these events are mutually exclusive, so P(red or black) = P(red) + P(black) = ½ + ½ = 1.
互斥事件不能同时发生。对于这样的事件 A 和 B，P(A 或 B) = P(A) + P(B)。这是互斥事件的加法法则。从一副标准纸牌中抽到红牌或黑牌：这些事件互斥，因此 P(红或黑) = P(红) + P(黑) = ½ + ½ = 1。
events are not mutually exclusive, you must subtract the overlap: P(A or B) = P(A) + P(B) – neral addition rule, though at Extended level, you will often use this with Venn diagrams.
当事件不互斥时，必须减去重叠部分：P(A 或 B) = P(A) + P(B) – P(A 与 B)。这是一般加法法则，不过在拓展层级，你常会结合维恩图使用该法则。
4. Independent Events and the Multiplication Rule | 独立事件与乘法法则
Independent events do not affect each other' then rolling an odd number on a fair die: the two rolls are independent, so P(6 and odd) = ⅙ × ½ = ¹⁄₁₂.
独立事件互不影响对方的概率。对于独立事件 A 和 B，P(A 与 B) = P(A) × P(B)。掷一枚公平骰子，先掷出 6 再掷出奇数：两次投掷独立，因此 P(6 与奇数) = ⅙ × ½ = ¹⁄₁₂。
Be careful: in exam questions, independence is often stated or implied by the phrase 'with replacement'. If items are selected with replacement, the trials remain independent because the probabilities do not change.
请注意：在考题中，独立性常会明确说明，或通过“有放回”一词暗示。如果抽取物品是有放回的，试验保持独立，因为概率不变。
5. Conditional Probability and 'Given That' | 条件概率与“给定……条件下”
the probability of event A occurring given that event B has already occurred, written P(A|B). The formula is P(A| (A and B) / P(B). This topic appears mainly in Extended tier. Crucially, you interpret the 'given that' statement to restrict the sample space.
条件概率是指在事件 B 已经发生的条件下事件 A 发生的概率，记作 P(A|B)。公式为 P(A|B) = P(A 与 B) / P(B)。该主题主要在拓展层级出现。关键是要将“给定……”表述理解为限制样本空间。
3 red and 5 blue marbles. Two marbles are drawn without replacement. Find P(second is red | nd red) = (5/8) × (3/7) = 15/56. Divided by P(first blue)=5/8 gives (15/56) ÷ (5/8) = 3/7.
示例：一个袋子里有 3 颗红色和 5 颗蓝色弹珠。无放回地抽取两次。求 P(第二颗红 | 第一颗蓝)。若第一颗是蓝色，剩余：3 红，4 蓝。所以 P(第二颗红) = 3/7。公式验证：P(第一颗蓝第二颗红) = (5/8) × (3/7) = 15/56。除以 P(第一颗蓝)=5/8 得 (15/56) ÷ (5/8) = 3/7。
6. Tree Diagrams – With and Without Replacement | 树形图 – 有放回与无放回
Tree diagrams systematically display the outcomes of multi‑stage experiments. Probabilities are written on branches. Multiply along branches for 'and' probabilities; add different branch paths for 'or' probabilities. They are especially useful for conditional problems where probabilities change after the first event.
树形图系统地展示多阶段实验的结果。概率写在树枝上。沿树枝相乘求得“与”的概率；将不同分支路径相加求得“或”的概率。当第一事件后概率发生变化时，它们对条件问题特别有用。
Without replacement: the probabilities on the second set of branches depend on the first outcome. Always check that the sum of probabilities from a single point equals 1. For a bag with 4 red and 3 green sweets, pick two without replacement. The tree clearly shows outcomes: RR, RG, GR, GG, with correct path probabilities.
无放回：第二组树枝上的概率取决于第一次结果。始终检查从同一点出发的概率之和等于 1。对于一个装有 4 颗红色和 3 颗绿色糖果的袋子，无放回地取两次。树形图清晰显示 RR、RG、GR、GG 结果及正确的路径概率。
With replacement, probabilities on second branches remain identical to the first stage because the composition is restored. Whether to use multiplication or a tree depends on the question’s demand for clarity.
有放回时，第二级树枝上的概率与第一阶段相同，因为组合已恢复。是直接相乘还是画树形图取决于题目对清晰性的要求。
7. Venn Diagrams and Set Notation in Probability | 维恩图与概率中的集合符号
Venn diagrams illustrate sets and their relationships, extremely handy for solving probability problems involving overlap. The universal set E represents the whole sample space. You typically place probabilities (or numbers of outcomes) inside regions. Common symbols: union A ∪ B (or), intersection A ∩ B (and), complement A'.
维恩图展示集合及其关系，对解决涉及重叠的概率问题非常实用。全集 E 代表整个样本空间。你通常在各区域内放置概率（或结果数量）。常见符号：并集 A ∪ B（或），交集 A ∩ B（与），补集 A'。
For mutually exclusive events, the circles do not overlap, so P(A ∩ B) = 0. When given a Venn diagram, read probabilities carefully: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Also, remember P(A' ∪ B') = 1 – P(A ∩ B), which is a frequent trick in exams.
对于互斥事件，圆圈不重叠，因此 P(A ∩ B) = 0。给出维恩图时，仔细读取概率：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。此外，记住 P(A' ∪ B') = 1 – P(A ∩ B)，这是考试中常见的技巧。
u may be asked to shade regions or find conditional probabilities from a Venn diagram: P(A|B) = n(A ∩ B) / n(B) when using frequencies, or equivalently using probabilities.
你可能需要给区域涂色，或从维恩图中求条件概率：使用频数时 P(A|B) = n(A ∩ B) / n(B)，或用概率等价地表达。
8. Relative Frequency and Expected Frequency | 相对频率与期望频数
times event occurs) / (total number of trials). Over many trials, the relative frequency tends to stabilise around the theoretical probability, but it is not exact for a finite number of experiments.
相对频率是基于实验数据对概率的估计：相对频率 = (事件发生次数) / (总试验次数)。经过大量试验，相对频率有稳定在理论概率附近的趋势，但对于有限次实验并不精确。
es the expected number of heads is 0.4 × 200 = 80. This does not guarantee exactly 80 heads, but represents the average over many repetitions.
期望频数由 (理论概率) × (试验次数) 计算得出。若一枚不均匀硬币正面概率为 0.4，则在 200 次抛掷中正面的期望次数为 0.4 × 200 = 80。这不保证正好 80 次正面，但代表多次重复的平均值。
9. Combined Events and Systematic Strategies | 组合事件与系统性策略
For combined events like 'rolling two dice and summing the scores' listing outcomes in a table is efficient. The sum of 7 has the highest probability: 6/36 = ⅙. Students often confuse the event 'score of 3 or 4' with the outcome of a single die. Use clear separating language.
对于“掷两枚骰子求总分”的组合事件，用表格列出结果很高效。总和为 7 的概率最高：6/36 = ⅙。学生常混淆“得分为 3 或 4”的事件和单颗骰子的结果。应使用明确分隔的语言。
Another common trap is thinking that getting a head and a tail in two coin tosses is a single outcome, but there are two distinct orders (HT and TH). Always distinguish ordered outcomes from unordered events when calculating probabilities.
另一个常见陷阱是认为抛两枚硬币得到一正一反是一个结果，但实际有两种不同顺序（HT 和 TH）。在计算概率时，永远要将有序结果与无序事件区分开来。
10. Exam Tips and Common Misconceptions | 考试技巧与常见误解
Always check that your probabilities are within 0 and 1. Probabilities given as fractions should be simplified unless the question specifies otherwise. For 'and' probabilities, multiply along branches only after confirming independence or applying the conditional rule correctly.
始终检查概率值是否在 0 与 1 之间。以分数给出的概率应化简，除非题目另有说明。针对“与”的概率，仅在确认独立或正确应用条件规则后才沿树枝相乘。
When using a tree diagram, label the ends with events, not just raw probabilities. This reduces mistakes when adding paths. If a question asks for 'at least one', consider using 1 – P(none). This is often quicker and avoids missing cases.
使用树形图时，在末端标出事件，而不仅仅是概率。这能减少相加路径时的错误。若题目要求“至少一个”，考虑使用 1 – P(一个也没有)。这往往更快，且能避免遗漏情况。
In conditional probability questions, underline the 'given that' part to consistently restrict the sample space. Many errors arise from dividing by the wrong total. Finally, cross‑check your answer with common sense; if you expect a probability around ½ but get 0.02, review your working.
在条件概率问题中，在“给定……”部分划线，以便始终限制样本空间。许多错误源于除以错误的总数。最后，用常识交叉检查你的答案；如果你预期概率约为 ½ 却得到 0.02，请复查你的解答过程。
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