Combined Events and Probability | 组合事件与概率

📚 Combined Events and Probability | 组合事件与概率

In A-Level Mathematics under Edexcel, the study of probability extends to situations where multiple events occur together. These are known as combined events. Understanding how to calculate probabilities for combined events — whether they are mutually exclusive, independent, or conditional — is essential for success in the Statistics component. This article explores the core principles, rules, diagrams, and common pitfalls associated with combined events.

在 Edexcel A-Level 数学的统计部分，概率的学习延伸至多个事件同时发生的情形，这些被称为组合事件。掌握如何计算组合事件的概率——无论它们是互斥、独立还是有条件——对考试成功至关重要。本文将深入探讨组合事件的核心原理、法则、示意图以及常见陷阱。

1. Introduction to Combined Events | 组合事件简介

A combined event occurs when two or more events happen together, either simultaneously or in sequence. For example, rolling a die and flipping a coin simultaneously, or drawing two cards from a deck without replacement. The key is to determine the likelihood of outcomes that involve multiple random processes. In Edexcel A-Level, you are expected to handle combined events using set notation, Venn diagrams, tree diagrams, and probability formulas.

当两个或多个事件一起发生（同时或依次）时，就构成了组合事件。例如，同时掷一枚骰子和抛一枚硬币，或者从一副牌中不放回地抽取两张牌。关键在于确定涉及多个随机过程的结果的可能性。在 Edexcel A-Level 考试中，你需要能够运用集合符号、维恩图、树状图以及概率公式来处理组合事件。

Consider a fair coin and a fair six-sided die. The sample space for the combined experiment has 2 x 6 = 12 equally likely outcomes, such as (H,1), (H,2) … (T,6). The probability of getting a head and an even number is calculated by identifying the favourable outcomes: (H,2), (H,4), (H,6) — three outcomes. Hence, P(Head and Even) = 3/12 = 1/4.

考虑一枚公平硬币和一个公平六面骰子。该组合实验的样本空间有 2 x 6 = 12 个等可能结果，例如 (H,1)、(H,2) … (T,6)。计算得到正面且偶数的概率，可以通过找出有利结果：(H,2)、(H,4)、(H,6) — 三个结果。因此，P(正面且偶数) = 3/12 = 1/4。

2. Mutually Exclusive Events | 互斥事件

Two events A and B are mutually exclusive if they cannot occur at the same time. For example, when rolling a die, the events ‘roll an odd number’ and ‘roll a 2’ are mutually exclusive because a roll cannot be both odd and exactly 2. In set notation, A cap B = empty set. The probability of either A or B occurring is simply the sum of their individual probabilities:

如果两个事件 A 和 B 不能同时发生，则称它们互斥。例如，掷一枚骰子时，事件“掷出奇数”和“掷出 2”就是互斥的，因为掷出的点数不可能既是奇数又恰好为 2。用集合符号表示为 A cap B = 空集。A 或 B 发生的概率就是各自概率之和：

P(A union B) = P(A) + P(B)

P(A U B) = P(A) + P(B)

It is crucial to check for mutual exclusivity before adding probabilities. A common exam question gives a table of outcomes and asks whether events are mutually exclusive, requiring justification by checking if the intersection is empty.

在相加概率之前，必须先检查是否互斥，这一点至关重要。一道常见的考题会给出一个结果表格，并询问事件是否互斥，这需要通过验证交集是否为空来进行论证。

3. Non-Mutually Exclusive Events and the Addition Rule | 非互斥事件与加法法则

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