Probability Problems and Common Question Types | 概率问题与常见题型解析

📚 Probability Problems and Common Question Types | 概率问题与常见题型解析

Probability is a core topic in mathematics, appearing in both pure and applied contexts. This guide summarises the essential rules, common question types, and practical strategies for solving probability problems.

概率是数学中的核心考点，既出现在纯数学也出现在应用数学中。本文系统梳理概率的基本法则、常见题型以及实用解题策略。

1. Basic Terminology and Notation | 基本术语与符号

An experiment is a process that produces an outcome. A sample space is the set of all possible outcomes, often written as S. An event is a subset of the sample space, usually denoted by a capital letter such as A or B.

试验是一个产生结果的过程。样本空间是所有可能结果的集合，通常记为 S。事件是样本空间的一个子集，通常用大写字母如 A 或 B 表示。

The probability of an event A is written P(A). For equally likely outcomes, P(A) = number of favourable outcomes / total number of possible outcomes.

事件 A 的概率记为 P(A)。在等可能结果下，P(A) = 有利结果数 ÷ 可能结果总数。

Relative frequency is an experimental estimate of probability: it is calculated as the number of times an event occurs divided by the number of trials. The more trials you carry out, the closer the relative frequency tends to get to the true probability.

频率是概率的试验估计：用事件发生的次数除以试验总次数。试验次数越多，频率通常越接近真实概率。

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2. The Probability Scale and Sample Space | 概率范围与样本空间

Probabilities are numbers between 0 and 1, inclusive. A probability of 0 means an event is impossible; a probability of 1 means it is certain.

概率是介于 0 和 1 之间的数，包含 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。

Listing the sample space explicitly is often the clearest first step. For example, when rolling two fair six-sided dice, the sample space has 36 equally likely outcomes.

明确列出样本空间通常是最清晰的第一步。例如，掷两枚均匀六面骰时，样本空间共有 36 个等可能结果。

A useful way to list outcomes is to use ordered pairs. For two dice, all pairs (a, b) where a and b each belong to {1, 2, 3, 4, 5, 6} form the complete sample space.

列结果时常用有序数对。对于两枚骰子，所有满足 a、b 都属于 {1, 2, 3, 4, 5, 6} 的数对 (a, b) 构成完整样本空间。

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3. Probability of Simple Events | 单一事件的概率

For a single event, direct counting gives the probability. If every outcome in the sample space is equally likely, then P(A) = |A| / |S|.

对于单一事件，直接计数即可得到概率。若样本空间中每个结果等可能，则 P(A) = |A| ÷ |S|。

P(A) = |A| / |S|

A common example is drawing a card from a standard 52-card deck. The probability of drawing a heart is 13/52 = 1/4.

常见例子是从一副 52 张扑克牌中抽牌。抽到红心的概率为 13/52 = 1/4。

Another classic example is rolling a fair six-sided die. The probability of rolling a number greater than 4 is 2/6 = 1/3, because the favourable outcomes are 5 and 6.

另一个经典例子是掷一枚均匀六面骰。掷出的点数大于 4 的概率为 2/6 = 1/3，因为有利结果为 5 和 6。

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4. Complementary Events | 对立事件

Every event A has a complement A‘, consisting of all outcomes not in A. The key rule is P(A) + P(A‘) = 1.

每个事件 A 都有对立事件 A‘，由所有不在 A 中的结果组成。核心法则为 P(A) + P(A‘) = 1。

P(A) + P(A’) = 1

Using the complement is particularly helpful when the desired event is complicated but its complement is simple. For example, “at least one head” in three coin tosses has complement “no heads”.

当所求事件复杂而对立事件简单时，利用补事件特别有效。例如，三次掷硬币中“至少一次正面”的补事件是“没有正面”。

The probability of no heads in three tosses is (1/2)³ = 1/8. Therefore, the probability of at least one head is 1 − 1/8 = 7/8.

三次都没有正面的概率为 (1/2)³ = 1/8。因此，至少出现一次正面的概率为 1 − 1/8 = 7/8。

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5. Mutually Exclusive Events and Addition Rule | 互斥事件与加法法则

Two events are mutually exclusive if they cannot happen at the same time. For such events, P(A or B) = P(A) + P(B).

两个事件互斥是指它们不能同时发生。对于互斥事件，P(A 或 B) = P(A) + P(B)。

If the events are not mutually exclusive, the general addition rule is P(A or B) = P(A) + P(B) − P(A and B).

若事件并不互斥，一般加法法则为 P(A 或 B) = P(A) + P(B) − P(A 且 B)。

P(A or B) = P(A) + P(B) − P(A and B)

For example, when selecting a student at random, suppose P(studies French) = 0.3 and P(studies Spanish) = 0.4. If these are mutually exclusive, then P(studies at least one of French or Spanish) = 0.3 + 0.4 = 0.7.

例如，随机选一名学生，设其学法语的概率为 0.3，学西班牙语的概率为 0.4。若两者互斥，则至少学其中一门外语的概率为 0.3 + 0.4 = 0.7。

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6. Independent Events and Multiplication Rule | 独立事件与乘法法则

Two events are independent if the occurrence of one does not affect the probability of the other. For independent events, P(A and B) = P(A) × P(B).

两个事件独立是指一个事件的发生不影响另一个事件的概率。对于独立事件，P(A 且 B) = P(A) × P(B)。

P(A and B) = P(A) × P(B)

For example, if the probability that a student passes mathematics is 0.8 and passes physics is 0.7, and the two results are independent, then the probability of passing both is 0.8 × 0.7 = 0.56.

例如，若某学生数学及格的概率为 0.8，物理及格的概率为 0.7，且两科成绩独立，则两科都及格的概率为 0.8 × 0.7 = 0.56。

Independence is often assumed in repeated trials, such as tossing coins or rolling dice. However, when drawing without replacement, outcomes are not independent.

独立常在重复试验中被假设，例如掷硬币或掷骰子。但若采用不放回抽取，结果就不再独立。

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7. Conditional Probability | 条件概率

Conditional probability measures the probability of one event given that another event has occurred. It is written P(A | B) and calculated as P(A | B) = P(A and B) / P(B), where P(B) > 0.

条件概率度量的是在另一事件已经发生的条件下某事件发生的概率。记为 P(A | B)，计算公式为 P(A | B) = P(A 且 B) ÷ P(B)，其中 P(B) > 0。

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

A classic example: A bag contains 3 red and 2 blue marbles. If the first marble drawn is red and not replaced, the probability that the second marble is blue is 2/4 = 1/2.

经典例子：袋中有 3 个红球和 2 个蓝球。若第一次取出红球且不放回，则第二次取出蓝球的概率为 2/4 = 1/2。

When using conditional probability, always check whether a change has occurred in the sample space. For example, without replacement, the total number of marbles decreases after the first draw.

使用条件概率时，务必检查样本空间是否发生了改变。例如在不放回抽样中，第一次抽取后总球数会减少。

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8. Tree Diagrams | 树状图

Tree diagrams display the outcomes of successive events. Each branch represents a possible outcome, and probabilities along adjacent branches are multiplied.

树状图展示连续事件的结果。每个分支代表一种可能结果，相邻分支上的概率相乘。

When drawing a tree, label each branch with its probability and multiply along the paths to find combined probabilities. Add the probabilities of relevant paths to answer questions like “at least one”.

绘制树状图时，在每条分支上标出概率，并沿路径相乘得到联合概率。将相关路径的概率相加即可回答“至少一次”这类问题。

For example, suppose a multiple-choice question has two independent parts: the probability of getting the first part correct is 0.6, and the second part correct is 0.5. The tree has four paths. The probability of getting at least one correct is 0.6 × 0.5 + 0.6 × 0.5 + 0.4 × 0.5 = 0.3 + 0.3 + 0.2 = 0.8.

例如，某道选择题有两个独立小题：第一题答对的概率为 0.6，第二题答对的概率为 0.5。树状图共有四条路径。至少答对一题的概率为 0.6 × 0.5 + 0.6 × 0.5 + 0.4 × 0.5 = 0.3 + 0.3 + 0.2 = 0.8。

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9. Venn Diagrams and Two-Way Tables | 文氏图与双向表

Venn diagrams and two-way tables organise probabilities for overlapping events. They help identify intersections, unions, and complements.

文氏图和双向表用于整理重叠事件的概率，帮助识别交集、并集和补集。

In a Venn diagram, the intersection A ∩ B represents “A and B”; the union A ∪ B represents “A or B”. Two-way tables allow direct counting from given frequencies.

在文氏图中，交集 A ∩ B 表示“A 且 B”；并集 A ∪ BPublished by TutorHao | Mathematics Revision Series | aleveler.com

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