📚 Probability 6.1: Understanding Outcomes and Sample Spaces | 概率 6.1：理解结果与样本空间
Probability is a branch of mathematics that allows us to measure how likely an event is to happen. In this chapter, you will learn to describe outcomes, list all possible results of an experiment, and calculate basic probabilities using fractions. Understanding probability helps us make sense of chance in games, weather forecasts, and everyday decisions.
概率是数学的一个分支，使我们能够衡量一个事件发生的可能性有多大。在本章中，你将学习如何描述结果，列出一个实验的所有可能结果，并使用分数计算基本概率。理解概率有助于我们理解游戏、天气预报和日常决策中的随机性。
1. What is Probability? | 什么是概率？
Probability is a number that describes the chance of an event occurring. It ranges from 0 (impossible) to 1 (certain). In many real-life situations, we use the idea of probability to decide whether to carry an umbrella, play a board game, or even invest money. Mathematically, it is the foundation of statistics and risk analysis.
概率是一个描述事件发生机会大小的数字，范围从 0（不可能发生）到 1（必然发生）。在许多实际生活场景中，我们利用概率的概念来决定是否带伞、玩棋盘游戏，甚至投资。在数学上，概率是统计和风险分析的基础。
2. Random Experiment, Outcomes and Sample Space | 随机实验、结果与样本空间
A random experiment is a process where the result cannot be predicted with certainty before it happens. Each individual possible result is called an outcome, and the set of all possible outcomes is the sample space. For example, when tossing a fair coin, the sample space is {Heads, Tails}.
随机实验是一个在发生前无法确定预测结果的过程。每个单独的可能结果称为一个结果，而所有可能结果的集合就是样本空间。例如，抛掷一枚均匀硬币时，样本空间是 {正面, 反面}。
this as S = {1, 2, 3, 4, 5, 6}. If the die is fair, each outcome has the same probability.
当你掷一枚均匀的六面骰子一次时，样本空间由数字 1、2、3、4、5 和 6 组成。我们通常将其写为 S = {1, 2, 3, 4, 5, 6}。如果骰子是均匀的，每个结果出现的概率相同。
3. Equally Likely Outcomes | 等可能结果
Outcomes are equally likely if each one has the same chance of occurring. A fair coin and a fair die give equally likely outcomes. However, if a spinner is not balanced or a die is weighted, the outcomes are not equally likely. Most probability calculations at this stage assume fairness, so using the fraction of favourable outcomes over total outcomes is correct.
如果每个结果发生的可能性都相同，那么它们就是等可能的。一枚均匀硬币和一枚均匀骰子产生等可能结果。但如果转盘不平衡或骰子被加重，结果就不是等可能的。这个阶段的大多数概率计算都假设公平性，因此使用有利结果数除以总结果数的方法是正确的。
P(E) = number of favourable outcomes / total number of outcomes. For example, the outcomes 'win', 'lose' or 'draw' in a football match are not equally likely because the teams' abilities differ.
在使用公式 P(E) = 有利结果数 ÷ 总结果数之前，检查结果是否等可能非常重要。例如，在一场足球比赛中的结果“胜”、“负”或“平局”并不是等可能的，因为两队实力不同。
4. Probability of an Event | 事件的概率
An event is a specific set of outcomes we are interested in. For a single roll of a die, the event 'rolling an even number' includes the outcomes {2, 4, 6}. If all outcomes are equally likely, the probability of this event is:
一个事件是我们感兴趣的特定结果集合。对于一次掷骰子，“掷出偶数”这个事件包括结果 {2, 4, 6}。如果所有结果等可能，该事件的概率为：
P(even) = 3 / 6 = 1/2
We always simplify fractions when writing probabilities, as 1/2 is clearer than 3/6. A probability can also be written as a decimal (0.5) or a percentage (50%). The KS3 curriculum expects you to be comfortable switching between fractions, decimals and percentages.
我们在写概率时总要约简分数，因为 1/2 比 3/6 更清晰。概率也可以写成小数 (0.5) 或百分数 (50%)。KS3 课程要求你能够在分数、小数和百分数之间灵活转换。
5. The Probability Scale | 概率标度
The probability scale is a number line from 0 to 1 that shows how likely events are. Events near 0 are very unlikely; events near 1 are very likely. We can mark words like impossible, unlikely, even chance, likely and certain on this scale. Even chance corresponds to a probability of 1/2 or 0.5.
概率标度是一条从 0 到 1 的数轴，显示事件发生的可能性。接近 0 的事件非常不可能发生；接近 1 的事件非常可能发生。我们可以在这条标度上标记诸如不可能、不太可能、均衡机会、很可能和必然等词语。均衡机会对应的概率是 1/2 或 0.5。
Being able to place everyday statements on the scale helps build intuition. For instance, 'The sun will rise tomorrow' has a probability close to 1, whereas 'Finding a pearl in every oyster' has a probability close to 0. In exam questions, you may be asked to estimate probabilities from data or mark events on a scale.
能够将日常语句放在标度上有助于培养直觉。例如，“明天太阳会升起”的概率接近 1，而“每个牡蛎里找到珍珠”的概率接近 0。在考试题目中，你可能需要根据数据估计概率或在标度上标记事件。
6. The Language of Likelihood | 可能性用语
We use words such as impossible, unlikely, even chance, likely and certain to describe probability without numbers. These terms are useful when exact probabilities are unknown or when we make quick comparisons. For example, picking a red counter from a bag with 9 red and 1 blue is likely, but not certain.
我们使用诸如不可能、不太可能、机会均等、很可能和必然这类词语来描述无需数字的概率。当精确概率未知或我们进行快速比较时，这些术语很有用。例如，从一个装有 9 个红棋子和 1 个蓝棋子的袋子中摸出一个红棋子是很可能的，但不是必然的。
and 1; certain = 1. This connection helps when estimating probabilities from experiments.
KS3 学生应将这些词语与数值区间联系起来：不可能 = 0；不太可能介于 0 和 1/2 之间；机会均等 = 1/2；很可能介于 1/2 和 1 之间；必然 = 1。这种联系在从实验估计概率时很有帮助。
7. Calculating Theoretical Probability | 计算理论概率
The theoretical probability of an event E is given by the formula:
事件 E 的理论概率由以下公式给出：
P(E) = Number of favourable outcomes ÷ Total number of possible outcomes
For example, a bag contains 4 red, 2 blue and 1 green ball. The total number of outcomes when picking one ball is 7. The probability of picking a blue ball is 2/7. This is a theoretical value because it is based on the perfect model of equally likely outcomes.
例如，一个袋子里装有 4 个红球、2 个蓝球和 1 个绿球。从中取一个球时，总结果数为 7。摸到蓝球的概率是 2/7。这是一个理论值，因为它基于等可能结果的理想模型。
blue) + P(green) = 4/7 + 2/7 + 1/7 = 7/7 = 1. This rule can be used to check your answers.
记住：样本空间中所有可能结果概率之和总是 1。在上面的例子中，P(红) + P(蓝) + P(绿) = 4/7 + 2/7 + 1/7 = 7/7 = 1。可以用这个规则来检查答案。
8. Experimental Probability and Relative Frequency | 实验概率与相对频率
When you cannot assume equally likely outcomes, you can estimate probability by doing an experiment. The experimental probability of an event is found using:
当你不能假设等可能结果时，你可以通过做实验来估计概率。一个事件的实验概率如下：
Experimental Probability = Number of times the event occurs ÷ Total number of trials
. The more trials you do, the closer the experimental probability usually gets to the theoretical probability (0.5). This idea is known as the law of large numbers.
这也被称为相对频率。如果你抛掷一枚硬币 100 次，得到 47 次正面，那么正面的实验概率是 47/100 = 0.47。你做的试验次数越多，实验概率通常会越接近理论概率 (0.5)。这个想法被称为大数定律。
In a KS3 investigation, you might roll a die 60 times and record outcomes in a frequency table. The relative frequency for each number should be close to 1/6 if the die is fair. Discrepancies help you discuss bias and randomness.
在 KS3 的探究活动中，你可能会掷骰子 60 次，并在频数表中记录结果。如果骰子是均匀的，每个数字的相对频率应接近 1/6。这些差异能帮助你讨论偏差和随机性。
9. Representing Sample Spaces – Lists and Two-Way Tables | 样本空间的表示——列表与双向表
When two things happen together, like rolling two dice or flipping a coin and spinning a spinner, it is useful to list all outcomes systematically. A simple list can be made, but a two-way table (or sample space diagram) is often clearer. For example, rolling a red die and a blue die gives 36 equally likely outcomes. The table shows each pair (red score, blue score).
当两件事情同时发生时，比如掷两个骰子，或者抛硬币和转一个转盘，系统地列出所有结果是很有用的。可以列一个简单的清单，但双向表（或样本空间图）往往更清晰。例如，掷一个红色骰子和一个蓝色骰子会产生 36 个等可能的结果。表格会显示每一对（红色点数，蓝色点数）。
ability that the sum of two dice is 7 can be found by counting pairs such as (1,6), (2,5), …, (6,1) – there are 6 favourable outcomes out of 36, giving P(sum=7) = 6/36 = 1/6.
通过双向表，你可以轻松地数出属于某个事件的结果个数。要求出两个骰子点数之和为 7 的概率，可以数出 (1,6)、(2,5)、……、(6,1) 这些数对——共有 6 个有利结果，总共 36 个结果，得出 P(和为7) = 6/36 = 1/6。
Another way to organise outcomes is a tree diagram, which is especially useful when listing outcomes in sequence. Each branch represents a possible result, and the complete set of branches shows all outcomes. At KS3, tree diagrams are usually drawn without probabilities on branches; they simply map out the sample space.
另一种组织结果的方法是树状图，当按顺序列出结果时特别有用。每一个分支代表一种可能的结果，所有分支合起来显示了全部结果。在 KS3 阶段，树状图通常不标注分支上的概率；它们只是描绘样本空间。
10. Complementary Events and ‘Not’ Probabilities | 互补事件与“非”概率
The complement of an event A is 'not A', which includes all outcomes not in A. Because the total probability of all outcomes is 1, we have:
事件 A 的互补事件是“非 A”，包括所有不在 A 中的结果。因为所有结果的总概率是 1，我们有：
P(not A) = 1 – P(A)
If the probability of rain tomorrow is 0.3, then the probability it does not rain is 1 – ly useful when counting favourable outcomes is difficult but the complement’s outcomes are easy to count.
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。当数出有利结果很困难，但互补事件的结果容易数时，这条简单的规则极其有用。
e probability of no sixes (which is 25/36) and then subtract from 1: P(at least one 6) = 1 – 25/36 = 11/36.
例如，要找出掷两个骰子至少出现一个 6 的概率，通常更容易先计算没有 6 的概率 (25/36)，然后从 1 中减去：P(至少一个6) = 1 – 25/36 = 11/36。
11. Common Misconceptions in Probability | 概率中的常见误区
One of the most common errors is the belief that after a streak of heads, tails is 'due' to occur. This is called the gambler's fallacy. In reality, if a coin is fair, each toss is independent, and the probability of heads remains 1/2 regardless of previous results.
最常见的错误之一是认为连续出现几次正面后，反面就“该出现了”。这被称为赌徒谬误。事实上，如果硬币是均匀的，每次抛掷都是独立的，正面的概率始终是 1/2，无论之前的结果如何。
Another misconception is adding probabilities when events are not mutually exclusive, but this topic is mainly for later stages. At KS3, it is important to check that the sum of all possible outcomes is 1 and to use fractions, not percentages or decimals, when comparing probabilities unless the question asks for a specific format.
另一个误区是在事件不互斥时将概率相加，但这个主题主要在后阶段讲解。在 KS3，重要的是要检查所有可能结果的总和为 1，并且除非题目要求特定格式，在比较概率时使用分数而不是百分数或小数。
12. Practice and Real-Life Applications | 练习与实际应用
Probability is used in designing fair games, predicting weather, quality control in factories, and modelling risks in finance and health. By practising with spinners, dice, cards and marbles, you develop a strong intuition for chance and are better prepared for the Cambridge Checkpoint test.
概率被用于设计公平游戏、预测天气、工厂的质量控制，以及模拟金融和健康领域的风险。通过用转盘、骰子、扑克牌和弹珠进行练习，你将培养出对机会的强烈直觉，并更好地准备剑桥 Checkpoint 考试。
Use past paper questions and online quizzes to practise listing sample spaces, calculating probabilities, and interpreting experimental data. Record your working clearly, and always simplify fractions to earn full marks. The skills you build here are the foundation for IGCSE and beyond.
使用历年考题和在线测验来练习列出样本空间、计算概率和解读实验数据。清晰地记录你的解答过程，并始终约简分数以获得满分。你在这里打下的技能基础将是 IGCSE 及更高阶段学习的基石。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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