Probability of Single Events | 单事件概率

📚 Probability of Single Events | 单事件概率

In many everyday situations — from flipping a coin to checking the weather forecast — we hear statements like “There is an even chance of rain” or “It’s unlikely to snow today.” These expressions refer to the likelihood of an event happening, which in mathematics we call probability. This article explores the key concepts of probability for single events, following the content commonly encountered around page 167 of the Cambridge Checkpoint Mathematics coursebook. By the end, you will be able to use the probability scale, calculate theoretical probabilities for equally likely outcomes, and understand the ideas of mutually exclusive and exhaustive events.

在许多日常情景中——从抛硬币到查看天气预报——我们都会听到诸如“下雨的可能性是均等的”或“今天不可能下雪”这样的说法。这些表述指的是事件发生的可能性，在数学中我们称之为概率。本文探讨单事件概率的关键概念，内容与剑桥 Checkpoint 数学教材第 167 页附近常见的知识点一致。学完本文后，你将能够使用概率尺度、计算等可能结果的理论概率，并理解互斥事件和穷举事件的概念。

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1. What Is Probability? | 什么是概率？

Probability is a measure of how likely an event is to occur. It takes a numerical value between 0 and 1 inclusive. An event that is impossible has a probability of 0, while an event that is certain to happen has a probability of 1. For all other events, the probability is some fraction or decimal between these two extremes. In words, we also describe probabilities using terms such as ‘impossible’, ‘unlikely’, ‘even chance’, ‘likely’ and ‘certain’.

概率是衡量一个事件发生可能性的度量。它的数值介于 0 和 1 之间（含 0 和 1）。不可能发生的事件概率为 0，而必然发生的事件概率为 1。对于其他所有事件，概率是介于两个极端之间的某个分数或小数。用文字描述时，我们也会使用“不可能”、“不太可能”、“均等机会”、“很可能”和“必然”等词汇。

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2. The Probability Scale | 概率尺度

The probability scale is a visual line from 0 to 1 on which we can place events to show their likelihood. For example, rolling a 7 on a standard six-sided dice is impossible, so it sits at 0. Rolling a number less than 7 is certain, so it sits at 1. Getting a head when flipping a fair coin lies exactly at 0.5, the midpoint. We can label the scale with fractions, decimals or percentages — all are acceptable in KS3 mathematics.

概率尺度是一条从 0 到 1 的视线，我们可以将事件放在上面以显示其可能性。例如，掷出一个标准六面骰子的点数 7 是不可能事件，因此它位于 0 处。掷出小于 7 的点数是必然事件，因此它位于 1 处。抛一枚均匀硬币得到正面恰好位于 0.5 的中点。我们可以用分数、小数或百分比来标注尺度——在 KS3 数学中这些都是可以接受的。

Probability Scale: 0 (impossible) — unlikely — even chance (0.5) — likely — 1 (certain)

概率尺度：0（不可能）— 不太可能 — 均等机会（0.5）— 很可能 — 1（必然）

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3. Notation and Basic Terms | 符号与基本术语

We commonly use the letter P to denote probability. For an event A, the probability of A occurring is written as P(A). The set of all possible outcomes of an experiment is called the sample space. Each individual result is an outcome, and an event is a collection of one or more outcomes. For a fair coin, the sample space is {Head, Tail}, and the event ‘getting a Head’ contains one outcome.

我们通常使用字母 P 来表示概率。对于事件 A，A 发生的概率写作 P(A)。一次试验的所有可能结果组成的集合叫做样本空间。每一个单独的结果是一个结局，而事件则是一个或多个结局的集合。对于一枚均匀硬币，样本空间为 {正面, 反面}，事件“得到正面”包含一个结局。

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4. Calculating Probability: Equally Likely Outcomes | 计算概率：等可能结果

When all outcomes in the sample space are equally likely, the probability of an event E is given by:

当样本空间中所有结果都是等可能时，事件 E 的概率由以下公式给出：

P(E) = Number of favourable outcomes ÷ Total number of possible outcomes

P(E) = 有利结局的数量 ÷ 可能结果的总数

For example, when rolling a fair six-sided dice, the probability of rolling a 3 is 1/6 because there is one favourable outcome (3) out of six possible outcomes. Similarly, the probability of rolling an even number is 3/6 = ½, since the favourable outcomes are 2, 4 and 6.

例如，掷一个均匀的六面骰子时，掷出 3 的概率是 1/6，因为有利结局（3）有 1 个，而可能结果共有 6 个。类似地，掷出偶数的概率是 3/6 = ½，因为有利结局为 2、4 和 6。

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5. Writing Probabilities in Different Forms | 以不同形式书写概率

Probabilities can be written as fractions, decimals or percentages. For example, a probability of ¼ can also be written as 0.25 or 25%. At KS3, you should be comfortable converting between these forms. A probability of 0.2 (or 1/5) means the event is unlikely but possible, while a probability of 0.8 means the event is quite likely. Always simplify fractions where possible, unless a question asks for a specific form.

概率可以用分数、小数或百分比表示。例如，¼ 的概率也可以写成 0.25 或 25%。在 KS3 阶段，你应该能熟练地在这些形式之间转换。0.2（或 1/5）的概率意味着事件不太可能发生但存在可能性，而 0.8 的概率意味着事件相当可能发生。除非题目要求特定的形式，否则应尽可能化简分数。

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6. Probability of an Event Not Happening | 事件不发生的概率

If the probability of an event happening is P(A), then the probability of the event not happening is 1 − P(A). We often denote the complement of A as A’ or ‘not A’. For example, if the probability of rain tomorrow is 0.3, the probability that it will not rain is 1 − 0.3 = 0.7. This rule is very useful because sometimes it is easier to find the probability of the complement first.

如果一个事件发生的概率为 P(A)，那么该事件不发生的概率就是 1 − P(A)。我们通常将 A 的补集记作 A’ 或“非 A”。例如，如果明天下雨的概率是 0.3，那么不下雨的概率就是 1 − 0.3 = 0.7。这条规则非常有用，因为有时先求对立事件的概率会更容易。

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7. Mutually Exclusive Events | 互斥事件

Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a dice, the events ‘rolling a 2’ and ‘rolling an odd number’ are mutually exclusive because a single roll cannot be both 2 and odd. In contrast, ‘rolling an even number’ and ‘rolling a number greater than 4’ are not mutually exclusive, since rolling a 6 satisfies both. When events A and B are mutually exclusive, the probability that either A or B occurs is simply the sum of their individual probabilities: P(A or B) = P(A) + P(B).

如果两个事件不可能同时发生，则它们是互斥事件。例如，掷骰子时，事件“掷出 2”和“掷出奇数”是互斥的，因为一次投掷不可能同时是 2 和奇数。相反，“掷出偶数”和“掷出大于 4 的数”就不是互斥的，因为掷出 6 同时满足两者。当事件 A 与 B 互斥时，A 或 B 发生的概率就是它们各自概率之和：P(A 或 B) = P(A) + P(B)。

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8. Exhaustive Events | 穷举事件

A set of events is exhaustive if they cover all possible outcomes of the experiment. For instance, when flipping a coin, the events ‘Head’ and ‘Tail’ are exhaustive because one of them must occur. Similarly, the events ‘rolling a number less than 3’, ‘rolling a 3 or 4’ and ‘rolling a 5 or 6’ are exhaustive for a dice roll. When events are both mutually exclusive and exhaustive, the sum of their probabilities is exactly 1.

如果一组事件涵盖了试验的所有可能结果，那么它们就是穷举事件。例如，抛硬币时，事件“正面”和“反面”就是穷举事件，因为二者必居其一。同样，对掷骰子而言，事件“掷出小于 3 的数”、“掷出 3 或 4”以及“掷出 5 或 6”也是穷举的。当事件既互斥又穷举时，它们的概率之和恰好为 1。

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9. Sum of Probabilities of All Outcomes | 所有结果的概率之和

In any probability experiment, the sum of the probabilities of all possible distinct outcomes is always 1. This is a fundamental rule. For a fair dice, each of the six outcomes has a probability of 1/6, and 6 × (1/6) = 1. If you are given some probabilities in a question and they do not sum to 1, you may need to find a missing probability. For example, if a biased spinner has probabilities for red, blue and green as 0.2, 0.5 and x, then 0.2 + 0.5 + x = 1, giving x = 0.3.

在任何概率试验中，所有可能的互异结果的概率之和始终为 1。这是一条基本规则。对于均匀骰子，六个结果中每一个的概率都是 1/6，且 6 × (1/6) = 1。如果题目中给出的某些概率之和不为 1，你可能需要求出缺失的概率。例如，若一个非均匀转盘出现红、蓝、绿的概率分别为 0.2、0.5 和 x，则 0.2 + 0.5 + x = 1，从而 x = 0.3。

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10. Experimental Probability vs Theoretical Probability | 实验概率与理论概率

Theoretical probability is what we expect to happen based on equally likely outcomes, such as P(Head) = ½. Experimental probability, on the other hand, is based on the results of an actual experiment or trial. It is calculated as (number of times the event occurs) ÷ (total number of trials). As the number of trials increases, the experimental probability usually gets closer to the theoretical probability — this is known as the law of large numbers. At KS3, you will sometimes be asked to compare the two or to explain why they might differ for a small number of trials.

理论概率是基于等可能结果我们所期望发生的概率，例如 P(正面) = ½。而实验概率则基于实际实验或试行的结果，计算方法为（事件发生的次数）÷（总试行次数）。随着试行次数的增加，实验概率通常会越来越接近理论概率——这被称为大数定律。在 KS3 中，有时会要求你比较两者，或者解释为什么在试行次数较少时它们可能会不同。

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11. Solving Probability Word Problems | 解答概率文字题

When tackling word problems, first identify the sample space and list the favourable outcomes. Then apply the formula P(E) = favourable outcomes / total outcomes. Be careful with units — probability has no units. For mixed problems, draw a probability scale, a table, or a simple list to organise the information. For example: “A bag contains 3 red, 2 blue and 5 green counters. One counter is taken at random. What is the probability it is not blue?” Total counters = 3 + 2 + 5 = 10. Counters that are not blue = 3 + 5 = 8. So P(not blue) = 8/10 = 4/5.

解答文字题时，首先要确定样本空间并列出有利结局，然后应用公式 P(E) = 有利结局数 / 总结果数。注意单位——概率没有单位。对于混合题型，可以画出概率尺度、表格或简单的列表来组织信息。例如：“一个袋子里装有 3 个红色、2 个蓝色和 5 个绿色筹码。随机取出一个筹码，它不是蓝色的概率是多少？”筹码总数 = 3 + 2 + 5 = 10。非蓝色筹码 = 3 + 5 = 8。因此 P(非蓝色) = 8/10 = 4/5。

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12. Summary and Key Points | 总结与关键点

To master single‑event probability, remember these essentials: Probability is always a number from 0 to 1. Use the formula (favourable outcomes)/(total outcomes) for equally likely events. The probabilities of all mutually exclusive and exhaustive outcomes sum to 1. Use the complement rule 1 − P(A) to find the probability of ‘not A’. Recognise mutually exclusive and exhaustive sets of events. Convert between fractions, decimals and percentages fluently. With these tools, you can confidently solve the probability problems found around page 167 of your Cambridge Checkpoint coursebook and beyond.

要掌握单事件概率，请记住以下要点：概率始终是一个从 0 到 1 的数。对于等可能事件，使用公式（有利结局数）/（总结果数）。所有互斥且穷举的结果概率之和为 1。利用补集法则 1 − P(A) 来求“非 A”的概率。识别互斥事件组和穷举事件组。熟练地在分数、小数和百分比之间转换。有了这些工具，你就能自信地解决剑桥 Checkpoint 教材第 167 页附近以及更广泛范围内的概率问题。

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