Calculating Probabilities | 概率计算

📚 Calculating Probabilities | 概率计算

Probability is the branch of mathematics that quantifies uncertainty. In the Edexcel A Level Mathematics specification, calculating probabilities involves combining rules for mutually exclusive events, independent events, conditional probability, and counting techniques to solve real-world and abstract problems. This article revises the core methods and exam techniques you need.

概率是量化不确定性的数学分支。在 Edexcel A Level 数学考试大纲中，概率计算涉及结合互斥事件、独立事件、条件概率规则以及计数技巧，来解决现实和抽象问题。本文复习你需要掌握的核心方法和应试技巧。

1. Sample Space and Basic Probability | 样本空间与基本概率

A sample space is the set of all possible outcomes of an experiment. If all outcomes are equally likely, the probability of an event A is given by the number of favourable outcomes divided by the total number of outcomes.

样本空间是实验所有可能结果的集合。如果所有结果等可能，事件 A 的概率为有利结果数除以总结果数。

P(A) = favourable outcomes ÷ total outcomes

Probabilities always lie between 0 and 1 inclusive. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain.

概率始终在 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。

For any event A, the complement A’ satisfies P(A’) = 1 − P(A). This is useful when it is easier to find the probability that an event does not happen.

对于任何事件 A，其补事件 A’ 满足 P(A’) = 1 − P(A)。当更容易求出事件不发生的概率时，这个公式非常有用。

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2. Addition Rule and Mutually Exclusive Events | 加法法则与互斥事件

Two events A and B are mutually exclusive if they cannot happen at the same time. In this case, the intersection has probability zero: P(A ∩ B) = 0.

如果两个事件 A 和 B 不能同时发生，则它们互斥。此时交集的概率为零：P(A ∩ B) = 0。

For mutually exclusive events, the addition rule simplifies to P(A ∪ B) = P(A) + P(B).

对于互斥事件，加法法则简化为 P(A ∪ B) = P(A) + P(B)。

If events are not mutually exclusive, the general addition rule must be used: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

如果事件不是互斥的，则必须使用一般加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。

The intersection P(A ∩ B) is subtracted to avoid double-counting outcomes that belong to both events. For example, when selecting one card from a standard deck, the probability of drawing a heart or a king is not simply 13/52 + 4/52 because the king of hearts is counted twice.

减去交集 P(A ∩ B) 是为了避免重复计算同时属于两个事件的结果。例如，从一副标准扑克牌中抽取一张牌，抽到红桃或 K 的概率并不是简单的 13/52 + 4/52，因为红桃 K 被计算了两次。

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

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

如果两个事件 A 和 B 中一个的发生不影响另一个发生的概率，则它们独立。数学上，独立意味着 P(A ∩ B) = P(A) × P(B)。

Independence can be tested by checking whether P(A ∩ B) equals P(A)P(B). If the two values are not equal, the events are dependent.

独立性可以通过检验 P(A ∩ B) 是否等于 P(A)P(B) 来判断。如果两个值不相等，则事件是相关的。

P(A ∩ B) = P(A) × P(B)

For independent events, the probability that at least one of A or B occurs is P(A ∪ B) = P(A) + P(B) − P(A)P(B), or equivalently 1 − P(A’)P(B’).

对于独立事件，A 或 B 中至少一个发生的概率为 P(A ∪ B) = P(A) + P(B) − P(A)P(B)，或等价地 1 − P(A’)P(B’)。

This ‘at least one’ formula is often faster in exam questions where multiple independent events are considered.

在考虑多个独立事件的考试题中，“至少一个”公式通常更快。

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4. Conditional Probability | 条件概率

The conditional probability of A given B is denoted P(A | B) and is defined as P(A ∩ B) divided by P(B), provided P(B) > 0.

在 B 发生的条件下 A 的条件概率记作 P(A | B)，定义为 P(A ∩ B) 除以 P(B)，其中 P(B) > 0。

P(A | B) = P(A ∩ B) ÷ P(B)

Rearranging this formula gives the multiplication rule for dependent events: P(A ∩ B) = P(B) × P(A | B) = P(A) × P(B | A).

将该公式移项可得相关事件的乘法法则：P(A ∩ B) = P(B) × P(A | B) = P(A) × P(B | A)。

When A and B are independent, the conditional probability simplifies: P(A | B) = P(A) and P(B | A) = P(B). This is because B gives no extra information about A.

当 A 和 B 独立时，条件概率变得简单：P(A | B) = P(A) 且 P(B | A) = P(B)。这是因为 B 没有提供关于 A 的额外信息。

Exam questions often ask you to decide whether two events are independent or mutually exclusive from a given context or table, so you must know both definitions precisely.

考试题通常要求你根据给定情境或表格判断两个事件是独立还是互斥，因此你必须准确掌握两者的定义。

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5. Tree Diagrams | 树状图

Tree diagrams are useful for modelling multi-stage experiments, especially when outcomes are not independent and selections are made without replacement.

树状图适用于建模多阶段实验，尤其是当结果不独立且抽样为无放回时。

Each branch shows a conditional probability given the previous outcome. To find the probability of a particular path, multiply the probabilities along the branches. To find the probability of an event that can occur along several paths, add the path probabilities.

每条分支显示在前一结果下的条件概率。求某条路径的概率时，将沿分支的概率相乘。求一个可以通过多条路径发生的事件的概率时，将各路径概率相加。

Always label branches with probabilities that sum to 1 at each node, and write final outcomes at the ends. This helps you avoid missing branches or adding when you should multiply.

始终在每个节点标出总和为 1 的分支概率，并在末端写出最终结果。这有助于避免漏掉分支或在应当相乘时错误相加。

For example, a bag contains 3 red and 2 blue balls. Two balls are drawn without replacement. The probability of drawing two red balls is (3/5) × (2/4) = 3/10, because after the first red is removed, only 2 red balls remain out of 4.

例如，一个袋子里有 3 个红球和 2 个蓝球。无放回地抽取两次。抽到两个红球的概率为 (3/5) × (2/4) =

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