📚 int a-level maths worksheet conditional probability | A-Level 数学条件概率练习题知识点精讲
Conditional probability is a cornerstone of A-Level Mathematics, appearing in both the pure statistics and mechanics components. Many students find it tricky at first, but once you master tree diagrams, the formula P(A|B) = P(A ∩ B)/P(B), and the idea of updating probabilities based on new information, it becomes a powerful tool for solving real-world problems. This article breaks down typical conditional probability worksheet questions, covering key concepts, common mistakes, and exam-style worked examples, all designed to build your confidence and accuracy.
条件概率是A-Level数学的基石之一，常在统计与力学板块中出现。许多同学起初会觉得棘手，但一旦掌握了树状图、公式 P(A|B) = P(A ∩ B)/P(B) 以及根据新信息更新概率的思路，它就能成为解决实际问题的利器。本文拆解典型的条件概率练习题，涵盖核心概念、常见错误和真题风格范例，旨在帮你建立信心并提升准确度。
1. Defining Conditional Probability | 条件概率的定义
Conditional probability is the probability of an event occurring given that another event has already occurred. The notation P(A|B) reads “the probability of A given B”. The formula that defines it is:
条件概率是指在已知另一事件已发生的条件下，某事件发生的概率。记作 P(A|B)，读作“在B发生时A的概率”。其定义公式为：
P(A|B) = P(A ∩ B) / P(B), where P(B) > 0
This formula tells us that to find the probability of A happening under the condition that B has happened, we look at the fraction of B’s probability that is shared with A. It essentially restricts the sample space to B.
该公式告诉我们，在B已发生的条件下求A发生的概率，就是看在B的概率中，有多少部分是与A共有的。这本质上将样本空间限制在了B中。
In many worksheet problems, you will need to apply this formula directly. For example, if P(A ∩ B) = 0.2 and P(B) = 0.5, then P(A|B) = 0.2 / 0.5 = 0.4. But more commonly, you will calculate P(A ∩ B) using multiplication rules or tree diagrams before using the formula.
在众多练习题中，你需要直接套用该公式。比如，若 P(A ∩ B) = 0.2，P(B) = 0.5，则 P(A|B) = 0.2 / 0.5 = 0.4。但更多时候，你需要先用乘法规则或树状图算出 P(A ∩ B)，再代入公式。
A common mistake is confusing P(A|B) with P(B|A). These are very different unless A and B have equal probabilities. Always check which event is given.
常见错误是将 P(A|B) 与 P(B|A) 混淆。除非 A 与 B 概率相等，否则二者差异巨大。务必看清哪个事件是条件。
2. The Multiplication Rule | 乘法规则
Rearranging the conditional probability formula gives the multiplication rule, which is extremely useful for finding the probability of two events both occurring when the probability of one depends on the other.
将条件概率公式变形可得到乘法规则，这在求两个事件同时发生、且其中一个的概率依赖于另一个时极为有用。
P(A ∩ B) = P(A|B) × P(B) or P(A ∩ B) = P(B|A) × P(A)
This means that to find the intersection, you can multiply the conditional probability of A given B by the probability of B. You choose the version that matches the information you have.
这意味着求交集的概率，可用 A 在 B 条件下的概率乘以 B 的概率。你应根据已知信息选择合适的版本。
For example: A bag contains 6 red and 4 blue balls. Two balls are drawn without replacement. The probability the first is red is 6/10, and the probability the second is red given the first was red is 5/9. So the probability both are red is (6/10) × (5/9) = 30/90 = 1/3. This rule forms the backbone of tree diagram calculations.
例如：袋中有6红4蓝球，不放回地抽取两个。第一个为红的概率是 6/10，在第一个为红的条件下第二个为红的概率是 5/9。因此两个都是红色的概率为 (6/10) × (5/9) = 30/90 = 1/3。该规则是树状图计算的支柱。
When events are independent, P(A|B) = P(A), so the rule simplifies to P(A ∩ B) = P(A)P(B). Worksheets often ask you to determine if events are independent by comparing P(A|B) with P(A) or checking if P(A ∩ B) = P(A)P(B).
当事件相互独立时，P(A|B) = P(A)，乘法规则简化为 P(A ∩ B) = P(A)P(B)。练习题常要求你通过比较 P(A|B) 与 P(A)，或检验 P(A ∩ B) 是否等于 P(A)P(B) 来判断事件是否独立。
3. Tree Diagrams and Conditional Probability | 树状图与条件概率
Tree diagrams are the most common visual aid for solving conditional probability problems. They help you map out all possible outcomes and their probabilities in a structured manner, especially for sequential experiments like drawing cards or balls without replacement.
树状图是解决条件概率问题最常用的视觉工具。它能以结构化的方式展示所有可能的结果及其概率，尤其适用于不放回抽卡或抽球等有序试验。
In a tree diagram, each branch represents a possible outcome, and the probabilities on the secondary branches are conditional probabilities. For example, after a first red ball is drawn, the probabilities for the second draw change because the composition of the bag has changed.
在树状图中，每条枝条代表一个可能的结果，次级枝条上的概率均为条件概率。例如，抽出一个红球后，第二次抽取的概率会因袋中球数变化而改变。
To find the probability of a particular path (e.g., red then blue), you multiply the probabilities along the branches. To find the total probability of an event that can occur via multiple paths, you add the probabilities of those paths. Many conditional probability questions ask: “Given that the second ball is red, what is the probability the first was blue?” This reverses the sequence and requires careful use of the formula P(A|B) = P(A ∩ B) / P(B).
求某条路径的概率（如红后蓝），将沿途枝条概率相乘。若某事件可通过多条路径发生，则将这些路径的概率相加。很多条件概率题目会问：“已知第二个球是红色，求第一个球是蓝色的概率。”这颠倒了顺序，需谨慎使用公式 P(A|B) = P(A ∩ B) / P(B)。
Always label the branches clearly, write down both conditional and unconditional probabilities, and show your working step by step. Tree diagrams also help avoid the mistake of assuming events are independent when they are not.
务必清晰标注枝条，写下条件与非条件概率，并逐步展示计算过程。树状图还能避免在事件不独立时错误地假定其独立。
4. Working Out P(B) from a Tree: The Law of Total Probability | 从树状图中求 P(B)：全概率公式
In a typical worksheet question, you might be asked to find P(B) when B can occur after two different first-stage events A and A’. The law of total probability states:
在典型的练习题中，你可能需要求 B 的概率，而 B 可能发生在两个不同的第一阶段事件 A 与 A’ 之后。全概率公式如下：
P(B) = P(B|A) × P(A) + P(B|A’) × P(A’)
This formula extends to any number of mutually exclusive and exhaustive events. It simply sums all paths in the tree that lead to B.
该公式可推广到任意多个互斥且穷举的事件。它本质上就是将树状图中所有通往 B 的路径概率相加。
For instance, suppose the probability of raining on any day is 0.3. If it rains, the probability I am late for school is 0.8; if it does not rain, the probability I am late is 0.1. The overall probability I am late, P(L), is (0.8 × 0.3) + (0.1 × 0.7) = 0.24 + 0.07 = 0.31. This value is then used as the denominator when calculating conditional probabilities like P(rain | late).
例如，假设某天下雨的概率为 0.3。若下雨，我上学迟到的概率为 0.8；若不下雨，迟到的概率为 0.1。我迟到的总概率 P(L) = (0.8 × 0.3) + (0.1 × 0.7) = 0.24 + 0.07 = 0.31。该值随后会用作分母，计算如 P(下雨|迟到) 之类的条件概率。
Many students forget to use the total probability as the denominator and instead try to use a simple fraction. This is one of the most common errors. Always identify which event is ‘given’ and compute its overall probability from the tree.
许多学生忘记用全概率作分母，而试图用简单的分数，这是最常见的错误之一。务必先确定哪个事件是“条件”，并从树中算出它的总概率。
5. Using Venn Diagrams and Two-Way Tables | 使用维恩图和双向表
Conditional probability can also be explored through Venn diagrams and two-way tables. These are particularly helpful when you are given counts or probabilities of overlaps between sets, rather than sequential experiments.
条件概率也可通过维恩图和双向表来探索。当给的是集合间的重叠计数或概率，而非有序试验时，这些工具特别有用。
In a two-way table, you might see the number of students studying Maths and Physics. From the table, you can read off P(Maths ∩ Physics), P(Maths), and P(Physics). Then P(Physics | Maths) is the number in the intersection divided by the row or column total for Maths.
在双向表中，你可能会看到学习数学和物理的学生人数。由表可直接读出 P(数学 ∩ 物理)、P(数学) 和 P(物理)。那么 P(物理|数学) 就是交集人数除以数学所在行或列的总数。
Venn diagrams visually show the sample space and the overlap. The conditional probability P(A|B) corresponds to the ratio of the area of A ∩ B to the area of B. When given pure numbers, simply take the counts or probabilities in the overlapping region and divide by the total of the conditioning event.
维恩图直观地展示样本空间与交集。条件概率 P(A|B) 对应于 A ∩ B 的面积与 B 的面积之比。当给定数字时，只需将重叠区域的计数或概率除以条件事件的总数。
Worksheet problems often present data in a table and ask: “Given that a student studies Maths, find the probability they also study Physics.” Always check whether the table gives frequencies or probabilities, and adjust your calculation accordingly. If frequencies, convert to probabilities by dividing by the grand total first, or keep as frequencies and divide appropriately.
练习题常以表格呈现数据，并问：“已知某学生学习数学，求他也学习物理的概率。”务必检查表格给出的是频数还是概率，并相应调整计算。若是频数，可先除以总计转化为概率，或保持频数并恰当相除。
6. Bayes’ Theorem in Simple Settings | 简单情形下的贝叶斯定理
Many conditional probability worksheet questions are effectively an application of Bayes’ Theorem, though at A-Level you are not always required to quote the theorem formally. The idea is reversing a conditional probability.
尽管 A-Level 并不总要求正式引用贝叶斯定理，许多条件概率练习题实质上是贝叶斯定理的应用。其核心思路是反转条件概率。
Bayes’ theorem states: P(A|B) = [P(B|A) × P(A)] / P(B). The numerator is one path to B (via A), and the denominator is the sum of all paths to B (via A and via A’, etc.).
贝叶斯定理表达式为：P(A|B) = [P(B|A) × P(A)] / P(B)。分子是通向 B 的一条路径（经 A），分母是通向 B 的所有路径之和（经 A 与经 A’ 等）。
For example: A disease affects 1% of a population. A test for the disease is 95% accurate (sensitivity) and gives false positives 3% of the time. If a random person tests positive, what is the probability they actually have the disease? Here P(Disease) = 0.01, P(Pos|Disease) = 0.95, P(Pos|No Disease) = 0.03. Then P(Disease|Pos) = (0.95 × 0.01) / (0.95×0.01 + 0.03×0.99) = 0.0095 / (0.0095 + 0.0297) ≈ 0.242. This counter-intuitive result is a classic Bayes’ problem.
例如：某疾病在人群中的发病率为 1%。对该疾病的检测准确度为 95%（灵敏度），且有 3% 的假阳性率。若随机一人检测呈阳性，他实际患病的概率是多少？此处 P(患病) = 0.01，P(阳性|患病) = 0.95，P(阳性|未患病) = 0.03。于是 P(患病|阳性) = (0.95 × 0.01) / (0.95×0.01 + 0.03×0.99) = 0.0095 / (0.0095 + 0.0297) ≈ 0.242。这一反直觉的结果是经典的贝叶斯问题。
In the exam, you can answer such questions step by step using the conditional probability formula and the law of total probability without explicitly naming Bayes’ theorem. This is perfectly acceptable and often clearer.
在考试中，你可以用条件概率公式和全概率公式逐步作答，无需明确提及贝叶斯定理。这完全可接受，且往往更加清晰。
7. Independent Events and Conditional Probability | 独立事件与条件概率
Two events A and B are independent if the occurrence of one does not affect the probability of the other. This leads to two key equivalent conditions used in worksheets:
若两事件 A 与 B 相互独立，其中一个的发生不影响另一个的概率。这在练习题中产生两个关键的等价条件：
- P(A|B) = P(A)
- P(B|A) = P(B)
- P(A ∩ B) = P(A) × P(B)
You can use any of these to test for independence. If P(A|B) does not equal P(A), then A and B are dependent. Be careful: mutually exclusive events (which cannot happen together) are not independent (unless one has probability zero), because if one occurs, the probability of the other becomes zero.
你可任选其一检验独立性。若 P(A|B) 不等于 P(A)，则 A 与 B 不独立。请注意：互斥事件（不能同时发生）并非独立（除非概率为零），因为若一个发生，另一个的概率就变为零。
For example, if you roll a fair die, let A = {even number} and B = {number greater than 2}. P(A) = 1/2. The probability of A given B is 2/4 = 1/2, so they are independent. However, if A = {1} and B = {odd number}, P(A|B) = 1/3, not 1/6, so they are dependent.
例如，掷一枚均匀骰子，设 A = {偶数}，B = {大于2的数}。P(A) = 1/2。在 B 条件下 A 的概率为 2/4 = 1/2，故二者独立。但若 A = {1}，B = {奇数}，P(A|B) = 1/3，不等于 1/6，故相关。
Worksheet questions often give P(A), P(B), and P(A ∩ B) and ask to determine if A and B are independent. Simply compute P(A) × P(B) and compare it with P(A ∩ B). If equal, they are independent.
练习题常给出 P(A)、P(B) 和 P(A ∩ B)，并要求判断 A 与 B 是否独立。只需计算 P(A) × P(B) 并与 P(A ∩ B) 比较。若相等，则独立。
8. Conditional Probability in ‘Given That’ Word Problems | “已知…求…”文字题中的条件概率
Many exam questions frame conditional probability in real-world contexts: medical testing, quality control, weather, and games of chance. The key phrase is “given that” or “if it is known that”. You should immediately identify the conditioning event and the event whose probability you need.
许多考题将条件概率置于实际情境中：医学检测、质量控制、天气和机会游戏。关键词是“已知…”或“若已知…”。你应立即确定条件事件和要求概率的事件。
For example: In a factory, machines A and B produce 60% and 40% of items. 2% of A’s items are defective and 5% of B’s are defective. If an item is chosen at random and found defective, what is the probability it came from machine A?
例如：某工厂中，机器 A 和 B 分别生产 60% 和 40% 的产品。A 的产品有 2% 次品，B 的产品有 5% 次品。若随机抽检一件并发现为次品，求它来自机器 A 的概率。
First, draw a tree: first branch A (0.6) and B (0.4); second branch defective (0.02 from A, 0.05 from B). The path A-defective has probability 0.6 × 0.02 = 0.012; B-defective has 0.4 × 0.05 = 0.02. Total defective probability = 0.012 + 0.02 = 0.032. Then P(A|defective) = 0.012 / 0.032 = 0.375.
首先画树状图：第一级枝 A (0.6) 和 B (0.4)；第二级枝次品（A 下 0.02，B 下 0.05）。路径 A-次品概率为 0.6 × 0.02 = 0.012；B-次品为 0.4 × 0.05 = 0.02。次品总概率 = 0.012 + 0.02 = 0.032。于是 P(A|次品) = 0.012 / 0.032 = 0.375。
Always write a clear answer in context, e.g., “The probability that the item came from machine A given it is defective is 0.375.” This demonstrates understanding.
务必写出明确的语境答案，如：“已知该产品为次品，它来自机器 A 的概率为 0.375。”这体现了对题目的理解。
9. Combining Permutations/Combinations with Conditional Probability | 排列组合与条件概率的结合
Some advanced A-Level worksheets include conditional probability questions that require counting outcomes using permutations and combinations. For example, selecting a committee with conditions on gender or role, and then imposing an additional condition.
一些进阶的 A-Level 练习题将条件概率与排列组合计数结合起来。例如，根据性别或角色有条件地选择委员会，再施加额外条件。
Suppose a group has 5 men and 4 women. Two people are selected at random to form a subcommittee. Find the probability that both are women, given that at least one is a woman.
假设某小组有 5 男 4 女。随机选两人组成分委员会。求在已知至少有一名女性的情况下，两人均为女性的概率。
Total ways to choose 2 from 9 is C(9,2)=36. P(at least one woman) = 1 – P(both men) = 1 – C(5,2)/C(9,2) = 1 – 10/36 = 26/36. P(both women) = C(4,2)/36 = 6/36. Thus, P(both women | at least one woman) = (6/36) / (26/36) = 6/26 = 3/13. Notice how the denominator became the condition’s probability.
从 9 人中选 2 人的总方式数为 C(9,2)=36。P(至少一名女性) = 1 – P(均为男性) = 1 – C(5,2)/C(9,2) = 1 – 10/36 = 26/36。P(均为女性) = C(4,2)/36 = 6/36。因此，P(均为女性|至少一名女性) = (6/36) / (26/36) = 6/26 = 3/13。注意分母是条件的概率。
Always identify the restricted sample space created by the condition. In this case, the condition reduces the sample space from all 2-person selections to only those with at least one woman. Then just count the favorable outcomes within that restricted space.
务必识别条件所产生的限制样本空间。在此例中，条件将样本空间从所有二人组合缩减为至少含一名女性的组合。然后只需在该限制空间内计数有利结果。
10. Conditional Probability with Probability Distributions | 条件概率与概率分布
Conditional probability also appears in discrete random variables. You may be given a joint probability distribution and asked to find P(X = x | Y = y). This involves extracting the appropriate row or column from the table.
条件概率同样出现在离散随机变量中。给出联合概率分布后，你可能需要求 P(X = x | Y = y)。这需要从表中提取相应的行或列。
For instance, a table shows probabilities for X taking values 1,2 and Y taking values A,B. To find P(X=2 | Y=A), you divide the cell probability P(X=2 ∩ Y=A) by the marginal total P(Y=A). This is essentially the same formula as before.
例如，表格给出了 X 取值 1,2 与 Y 取值 A,B 的概率。求 P(X=2 | Y=A) 时，用单元格概率 P(X=2 ∩ Y=A) 除以边缘总计 P(Y=A)。这本质上仍是同一公式。
Sometimes you are given the conditional distribution and one marginal distribution and asked to complete the table. Use P(X=x ∩ Y=y) = P(X=x|Y=y) × P(Y=y) to fill in the joint probabilities, then sum to get other marginals.
有时题目给出条件分布和一个边缘分布，要求补全表格。使用 P(X=x ∩ Y=y) = P(X=x|Y=y) × P(Y=y) 填入联合概率，再求和得到其他边缘概率。
Worksheets may also test the concept of memoryless property of the geometric distribution: P(X > s + t | X > s) = P(X > t). This is a special conditional probability property you might need to apply.
练习题还可能考察几何分布的无记忆性：P(X > s + t | X > s) = P(X > t)。这是一条特殊的条件概率性质，可能需要你加以应用。
11. Common Mistakes and How to Avoid Them | 常见错误与如何避免
1. Confusing the conditioning event: Always write out P(A|B) and identify which is B. If the question says “given that the student is male”, then Male is the conditioning event (denominator).
1. 混淆条件事件：务必写出 P(A|B) 并确认哪个是 B。若题目说“已知该学生是男生”，则男生是条件事件（分母）。
2. Using P(B|A) instead of P(A|B): These can be very different. Double-check the wording.
2. 用 P(B|A) 代替 P(A|B)：二者可能截然不同。仔细核对题目措辞。
3. Forgetting to use total probability in denominator: When the condition is a second-stage event, you must sum all paths that lead to it, not just one path.
3. 忘记在分母中使用全概率：当条件为第二阶段事件时，必须汇总所有通往它的路径，而非仅仅一条。
4. Assuming events are independent without checking: Just because events are consecutive does not mean they are independent. In ‘without replacement’ questions, they are dependent. Always check if probability changes.
4. 未经核查就假设事件独立：事件相继发生并不意味它们独立。在“不放回”问题中，它们是相关的。务必检查概率是否变化。
5. Neglecting to adjust sample space: In counting problems, after the condition is applied, the denominator changes. Write down the reduced sample space explicitly.
5. 忽略调整样本空间：在计数问题中，施加条件后分母会变。请明确写出缩减后的样本空间。
6. Rounding too early: Keep probabilities as fractions or decimals with sufficient precision until the final answer to avoid rounding errors in complex Bayes problems.
6. 过早舍入：在复杂的贝叶斯问题中，应将概率保留为分数或足够精度的十进制数，直至最终答案，以避免舍入误差。
12. Practice Worksheet Strategies | 练习题应试策略
When tackling a conditional probability worksheet, adopt a systematic approach: (1) Read carefully and identify the given information and the event whose probability is required. (2) Decide if a tree diagram, table, or Venn diagram is more appropriate. (3) Label all branches or cells with probabilities, using given conditional probabilities directly. (4) Clearly write down the conditional probability formula with the correct numerator and denominator. (5) Compute the denominator using the law of total probability if necessary. (6) Simplify final answer and present it in context.
面对条件概率练习题时，采用系统方法： (1) 仔细读题，找出已知信息和所要求概率的事件。 (2) 判断树状图、表格还是维恩图更合适。 (3) 用所给条件概率为所有枝条或单元格标注概率。 (4) 清晰写出条件概率公式，配以正确的分子与分母。 (5) 必要时使用全概率公式计算分母。 (6) 简化最终答案并结合语境呈现。
Practice with a variety of problems: without replacement, medical testing, weather forecasting, and two-stage experiments. The more you practice, the more intuitive the process becomes. Past papers are excellent for seeing how examiners structure ‘given that’ questions.
练习多种题型：不放回抽样、医学检测、天气预报和两阶段试验。练习越多，过程就越直观。历年真题是了解考官如何编排“给定条件”题目的绝佳资料。
Remember, conditional probability is not just a topic—it’s a way of thinking. It trains you to update beliefs based on evidence, a skill that extends far beyond mathematics.
请记住，条件概率不仅是一个知识点，更是一种思维方式。它训练你根据证据更新信念，这项技能远不止于数学领域。
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