📚 Probability Basics | 概率基础
Probability is the branch of mathematics that describes how likely an event is to occur. It gives us a way to measure uncertainty and make predictions about the future. Whether we are rolling dice, flipping coins, or checking the weather forecast, probability helps us understand the chances of different outcomes happening. In Key Stage 3, we begin to explore the language of probability, learn how to calculate simple probabilities, and represent them on a scale from 0 to 1.
概率是数学中描述事件发生可能性的分支。它让我们能够衡量不确定性并对未来做出预测。无论是掷骰子、抛硬币还是查看天气预报，概率都能帮助我们理解不同结果发生的机会有多大。在 KS3 阶段，我们将开始探索概率的语言，学习如何计算简单的概率，并在 0 到 1 的尺度上表示它们。
1. What is Probability? | 什么是概率？
Probability is a number that expresses the chance of an event happening. It is always between 0 and 1, inclusive. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain. For example, when you flip a fair coin, there are two equally likely outcomes: heads or tails. The probability of getting heads is ½, or 0.5.
概率是一个表示事件发生可能性的数字，它总是在 0 到 1 之间（含 0 和 1）。概率为 0 表示事件不可能发生，概率为 1 表示事件一定发生。例如，当你抛一枚均匀的硬币时，有两种等可能的结果：正面或反面。得到正面的概率是 ½，即 0.5。
The word 'chance' is often used in everyday language to mean probability. Phrases like 'fifty-fifty', 'even money', or 'one in a million' all describe probabilities informally. Probability theory helps us turn these phrases into precise numbers.
日常语言中常用“机会”一词来表示概率。“五五开”、“均等赔率”或“百万分之一”这样的说法都在非正式地描述概率。概率论帮助我们将这些说法转化为精确的数字。
2. Experiments and Outcomes | 实验与结果
A probability experiment, or trial, is any action or process that leads to a result. Examples include rolling a die, spinning a spinner, or picking a card from a deck. Each possible result is called an outcome. When we roll a standard six-sided die, the outcomes are 1, 2, 3, 4, 5, and 6.
概率实验（或试验）是任何会产生结果的动作或过程。例如，掷骰子、转动转盘或从一副牌中抽一张牌。每一个可能的结果称为一个“结果”。当我们掷一个标准的六面骰子时，结果有 1、2、3、4、5 和 6。
It is important that the outcomes of a probability experiment are well-defined. If the die was weighted or the spinner was not balanced, the outcomes would not be equally likely, which makes calculating probability more complicated. In KS3, we mostly deal with fair experiments where all outcomes have the same chance.
重要的是，概率实验的结果必须定义明确。如果骰子被灌铅或者转盘不平衡，结果就不等可能了，这会使得概率计算更加复杂。在 KS3 阶段，我们主要处理所有结果机会均等的公平实验。
3. Sample Space | 样本空间
The sample space is the set of all possible outcomes of an experiment. We usually list the outcomes inside curly brackets { }. For a coin toss, the sample space is {Heads, Tails}. For a six-sided die, it is {1, 2, 3, 4, 5, 6}. For two coins, the sample space can be written as {HH, HT, TH, TT}, where H stands for heads and T for tails.
样本空间是一个实验所有可能结果的集合。我们通常用花括号 { } 列出结果。对于抛硬币，样本空间是 {正面, 反面}。对于一个六面骰子，样本空间是 {1, 2, 3, 4, 5, 6}。对于两枚硬币，样本空间可以写成 {HH, HT, TH, TT}，其中 H 代表正面，T 代表反面。
Listing the sample space systematically is a key skill. A common method is to use a table or a two-way grid to show combinations. This ensures that no outcome is missed and that we can count the total number of equally likely outcomes easily.
系统地列出样本空间是一项关键技能。常用的方法是使用表格或二维网格来显示组合。这样可以确保不遗漏任何结果，并能轻松计数等可能结果的总数。
4. Events | 事件
An event is a particular set of outcomes that we are interested in. For example, when rolling a die, the event 'getting an even number' is the set {2, 4, 6}. The event can be described using words, and then we find the corresponding outcomes from the sample space.
事件是我们感兴趣的特定结果的集合。例如，掷骰子时，“得到偶数”这个事件是集合 {2, 4, 6}。事件可以用文字描述，然后我们从样本空间中找到对应的结果。
al number of possible outcomes. This formula works only when all outcomes are equally likely. We always assume fair conditions unless stated otherwise.
要计算一个事件的概率，我们使用公式：概率 = 有利结果的数量 ÷ 可能结果的总数。该公式仅在所有结果等可能时才成立。除非另有说明，我们通常假设实验是公平的。
5. Calculating Probability | 计算概率
The basic rule for calculating probability is:
计算概率的基本规则是：
vent) = Number of ways the event can happen ÷ Total number of outcomes in the sample space
P(事件) = 事件发生的方式数 ÷ 样本空间中结果的总数
For instance, a bag contains 3 red marbles and 2 blue marbles. If you pick one marble at random, the probability of picking a red marble is 3/5, because there are 3 favourable outcomes out of 5 total marbles. The probability of picking a blue marble is 2/5.
例如，一个袋子里有 3 个红色弹珠和 2 个蓝色弹珠。如果你随机取出一个弹珠，取出红色弹珠的概率是 3/5，因为在总共 5 个弹珠中有 3 个有利结果。取出蓝色弹珠的概率是 2/5。
Probabilities can be written as fractions, decimals, or percentages. So the probability 3/5 can also be written as 0.6 or 60%. The fraction form is often preferred because it shows the exact ratio without rounding.
概率可以用分数、小数或百分比表示。因此概率 3/5 也可以写成 0.6 或 60%。分数形式通常更受青睐，因为它在不进行四舍五入的情况下显示了精确的比值。
6. Probability Scale | 概率尺度
The probability scale is a visual way to show how likely an event is. The scale runs from 0 (impossible) to 1 (certain). Halfway along the scale is 0.5, which represents an even chance. Events that are likely but not certain will have probabilities greater than 0.5, while unlikely events have probabilities less than 0.5.
概率尺度是一种直观展示事件可能性的方式。尺度从 0（不可能）到 1（一定）。尺度的中点是 0.5，表示均等的机会。可能但非必然发生的事件概率大于 0.5，而不太可能发生的事件概率小于 0.5。
| Probability in words | Numerical range | Example |
|---|---|---|
| Impossible | 0 | Rolling a 7 on a standard six-sided die |
| Unlikely | Less than 0.5 | Picking a particular card from a full deck (1/52) |
| Even chance | 0.5 | Getting heads when flipping a fair coin |
| Likely | Greater than 0.5 | The sun rising tomorrow |
| Certain | 1 | A day will have 24 hours |
Understanding the probability scale helps us to describe events accurately and to compare the likelihood of different events. It is also a foundation for more advanced topics like probability trees and expected frequency.
理解概率尺度有助于准确描述事件并比较不同事件的可能性。它也是学习树形图和期望频率等更高级主题的基础。
7. Complementary Events | 互补事件
the original event does NOT happen. If A is an event, then the complement is written as A' or 'not A' he A'. The probabilities of an event and its complement add up to 1. That is, P(A) + P(not A) = 1.
一个事件的互补事件是原事件不发生的事件。如果 A 是一个事件，那么它的互补事件记作 A' 或“非 A”。一个事件与其互补事件的概率之和为 1，即 P(A) + P(非 A) = 1。
is very useful because sometimes it is easier to calculate the probability of the complement and then subtract from 1.
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。这个规则非常有用，因为有时计算互补事件的概率再减掉 1 会更容易。
e events that cannot happen at the same time. For instance, when rolling a die, the events 'rolling a 1' and 'rolling a 6' tion of probabilities for mutually exclusive events works simply: P(A or B) = P(A) + P(B).
互斥事件是指不能同时发生的事件。例如，掷骰子时，“掷出 1”和“掷出 6”就是互斥事件。互斥事件的概率加法很简单：P(A 或 B) = P(A) + P(B)。
8. Relative Frequency | 相对频率
Relative frequency is an experimental way of estimating probability. If we repeat an experiment many times, the relative frequency of an event is the number of times the event occurs divided by the total number of trials. As the number of trials increases, the relative frequency tends to get closer to the theoretical probability. This is known as the Law of Large Numbers.
相对频率是一种通过实验来估计概率的方法。如果我们多次重复一个实验，一个事件的相对频率就是该事件发生的次数除以总实验次数。随着实验次数的增加，相对频率往往会趋近于理论概率。这就是大数定律。
t is not exactly 0.5, it is close. If we toss it 1000 times, we would expect the relative frequency to be even closer to 0.5.
例如，如果我们抛一枚硬币 100 次，得到 47 次正面，那么正面的相对频率是 47/100 = 0.47。虽然它不恰好是 0.5，但很接近。如果我们抛 1000 次，我们可以预期相对频率会更接近 0.5。
In KS3, students are often asked to compare experimental probabilities with theoretical ones and to analyse why differences occur. Fairness, sample size, and experimental conditions are key factors that can affect relative frequency.
在 KS3 阶段，学生经常被要求比较实验概率与理论概率，并分析产生差异的原因。公平性、样本大小和实验条件是可能影响相对频率的关键因素。
9. Tree Diagrams | 树形图
Tree diagrams are a systematic way to list all possible outcomes of two or more events. Each branch of the tree represents a possible outcome, and probabilities are written along the branches. Tree diagrams are especially useful when events occur in sequence, such as flipping two coins or picking items from a bag with replacement.
树形图是列出两个或更多事件所有可能结果的系统方法。树的每个分支代表一个可能的结果，概率写在分支上。树形图在事件相继发生时尤其有用，例如抛两枚硬币，或有放回地从袋中抽取物品。
ty of two heads is ½ × ½ = ¼. The sum of probabilities of all final outcomes must equal 1.
要找到结果组合的概率，我们将路径上的概率相乘。例如，对于两次硬币抛掷，两个正面的概率是 ½ × ½ = ¼。所有最终结果的概率之和必须等于 1。
By using tree diagrams, students can visualise compound events and avoid missing outcomes. This method also extends to probability without replacement, where branches change their probabilities because the sample space size changes.
通过使用树形图，学生可以直观地看到复合事件，避免遗漏结果。这种方法也扩展到无放回的概率问题中，此时由于样本空间大小发生变化，分支的概率也会改变。
10. Review and Summary | 复习与总结
In this chapter, we covered the fundamental ideas of probability that every KS3 student should know. The main points are:
在本章中，我们涵盖了 KS3 学生应该掌握的每一个概率基本概念。主要要点如下：
- Probability measures the chance of an event happening on a scale from 0 to 1.
- 概率用从 0 到 1 的尺度衡量事件发生的可能性。
- The sample space is the set of all possible outcomes.
- 样本空间是所有可能结果的集合。
- ally likely outcomes, P(event) = number of favourable outcomes ÷ total number of outcomes.
- 对于等可能结果，P(事件) = 有利结果数量 ÷ 结果总数。
- The complement rule states that P(not A) = 1 – P(A).
- 互补规则指出 P(非 A) = 1 – P(A)。
- Relative frequency from experiments can estimate probability and approaches the true probability with more trials.
- 实验得出的相对频率可以估计概率，并随着试验次数的增加趋近于真实概率。
- Tree diagrams help list outcomes and calculate probabilities for sequences of events.
- 树形图有助于列出结果并计算一系列事件的概率。
Mastering these basics will prepare you for more advanced topics in probability, such as independent and dependent events, expected value, and Venn diagrams. Always practice by writing out sample spaces carefully and using clear notation.
掌握这些基础知识将使你为更高级的概率主题做好准备，例如独立事件与相关事件、期望值以及文氏图。请始终通过仔细写出样本空间和使用清晰的符号来进行练习。
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