📚 Probability and Sample Spaces | 概率与样本空间
Understanding probability helps us make predictions about how likely events are to happen. In KS3 Mathematics, you will learn to describe chance using numbers between 0 and 1, carry out experiments to estimate probabilities, and list all possible outcomes systematically using sample space diagrams. This article covers the key concepts you need to master, including experimental and theoretical probability, mutually exclusive events, and expected frequency.
理解概率可以帮助我们预测事件发生的可能性有多大。在KS3阶段的数学中，你将学会用0到1之间的数字描述机会，通过实验来估计概率，并使用样本空间图系统地列出所有可能的结果。本文涵盖了你需要掌握的关键概念，包括实验概率与理论概率、互斥事件以及期望次数。
1. Introduction to Probability | 概率简介
Probability is the branch of mathematics that deals with the likelihood of events occurring. It is used in weather forecasting, games of chance, risk assessment, and many everyday decisions. A probability is always expressed as a number between 0 (impossible) and 1 (certain). It can be written as a fraction, decimal, or percentage.
概率是数学中处理事件发生可能性的一个分支，应用于天气预报、机会游戏、风险评估及许多日常决策中。概率始终用一个介于0（不可能）和1（必然）之间的数来表示，可以写成分数、小数或百分数。
2. Probability Scale | 概率尺度
The probability scale ranges from 0 to 1. An event that is impossible has a probability of 0, while an event that is certain has a probability of 1. If an outcome is just as likely to happen as it is not to happen, its probability is ½ or 0.5. Words such as ‘unlikely’, ‘even chance’, and ‘likely’ help to describe positions on the scale.
概率尺度从0到1。不可能的事件的概率为0，必然的事件概率为1。如果一种结果发生的可能性与不发生的可能性一样大，其概率为½或0.5。像“不太可能”、“机会均等”和“很可能”这样的词语有助于描述尺度上的位置。
3. Experimental Probability | 实验概率
Experimental probability is found by carrying out an experiment or trial and recording the outcomes. It is calculated as:
Experimental probability = Number of times the event occurs ÷ Total number of trials
. For example, if you flip a coin 100 times and get heads 47 times, the experimental probability of heads is 47/100 = 0.47.
实验概率是通过进行实验或试验并记录结果得到的。它的计算方法是：
实验概率 = 事件发生的次数 ÷ 试验总次数
。例如，如果你抛一枚硬币100次，得到47次正面，那么正面的实验概率为47/100 = 0.47。
4. Theoretical Probability | 理论概率
Theoretical probability is what we expect to happen mathematically, without actually doing an experiment. It is given by:
P(event) = Number of favourable outcomes ÷ Total number of possible outcomes
, provided all outcomes are equally likely. For a fair six-sided die, the probability of rolling a 3 is 1/6.
理论概率是我们从数学上期望发生的结果，而无需实际进行实验。它由下式给出：
P(事件) = 有利结果的数量 ÷ 可能结果的总数
，前提是所有结果的可能性相等。对于一颗均匀的六面骰子，掷出3点的概率是1/6。
5. Sample Space Diagrams | 样本空间图
A sample space is the set of all possible outcomes of an experiment. We can show sample spaces clearly using lists or diagrams. For example, when a fair coin is tossed, the sample space is {Heads, Tails}. When a fair die is rolled, the sample space is {1, 2, 3, 4, 5, 6}. For combined events, we often use tables.
样本空间是某个实验所有可能结果的集合。我们可以用列表或图形清晰地表示样本空间。例如，抛一枚均匀硬币时，样本空间是{正面, 反面}。掷一颗均匀骰子时，样本空间是{1, 2, 3, 4, 5, 6}。对于组合事件，我们经常使用表格。
6. Listing Outcomes Systematically | 系统列举可能结果
To avoid missing outcomes, you should list them in a logical order. When flipping two coins, the systematic list is: HH, HT, TH, TT. The total number of outcomes can also be found using the product rule: multiply the number of outcomes for each event. For flipping a coin and rolling a die, there are 2 × 6 = 12 possible outcomes.
为了避免遗漏结果，应该按照逻辑顺序列出它们。当抛两枚硬币时，系统列表是：HH, HT, TH, TT。结果的总数也可以利用乘法法则求得：即将每个事件的结果数相乘。对于抛一枚硬币并掷一颗骰子，总共有 2 × 6 = 12 种可能的结果。
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 die, getting a 2 and getting a 5 are mutually exclusive. For mutually exclusive events A and B, the probability of either A or B occurring is given by:
P(A or B) = P(A) + P(B)
. This is known as the addition rule for mutually exclusive events.
如果两个事件不可能同时发生，它们就是互斥的。例如，掷骰子时，得到2点和得到5点是互斥的。对于互斥事件A和B，A或B发生的概率由下式给出：
P(A 或 B) = P(A) + P(B)
。这被称为互斥事件的加法法则。
8. Probability of an Event Not Happening | 事件不发生的概率
The probability that an event does not happen is 1 minus the probability that it does happen. This can be written as:
P(not A) = 1 − P(A)
. For instance, if the probability of rain tomorrow is 0.3, the probability that it will not rain is 1 − 0.3 = 0.7.
一个事件不发生的概率等于1减去该事件发生的概率。这可以写成：
P(非A) = 1 − P(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。
9. Expected Frequency | 期望次数
If you repeat an experiment many times, the expected frequency of an event is the number of times you would expect it to occur. It is found by multiplying the probability by the number of trials:
Expected frequency = P(event) × number of trials
. For a die rolled 300 times, the expected number of sixes is (1/6) × 300 = 50.
如果你多次重复一个实验，一个事件的期望次数就是你预期它发生的次数。它由概率乘以试验次数得到：
期望次数 = P(事件) × 试验次数
。对于一颗掷300次的骰子，六点出现的期望次数为 (1/6) × 300 = 50。
10. Combined Events and Two-Way Tables | 组合事件与双向表
For combined events, such as rolling two dice, a two-way table (also known as a sample space table) helps visualise all outcomes. The table below shows the sum of two dice. From the table, you can find probabilities like P(sum = 7) = 6/36 = 1/6.
| + | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
对于组合事件，例如掷两颗骰子，双向表（也称样本空间表）有助于将所有的结果可视化。上表展示了两颗骰子的点数之和。通过这张表，你可以求出如 P(和为7) = 6/36 = 1/6 这样的概率。
11. Comparing Experimental and Theoretical Probability | 比较实验概率与理论概率
As the number of trials increases, the experimental probability usually gets closer to the theoretical probability. This is often called the ‘law of large numbers’. For a biased spinner, however, the experimental probability may remain quite different from a simple theoretical prediction, which signals that the outcomes are not equally likely.
随着试验次数的增加，实验概率通常会越来越接近理论概率，这常被称为“大数定律”。然而，如果是一个不均匀的转盘，其实验概率可能一直与简单的理论预测相差较大，这暗示着结果并不是等可能的。
12. Summary and Key Points | 总结与要点
Probability is measured on a scale from 0 to 1. The theoretical probability uses equally likely outcomes, while experimental probability comes from data. Sample space diagrams and systematic listing guarantee you count all outcomes. Remember the addition rule for mutually exclusive events and the complement rule for an event not happening. Practise finding expected frequencies to connect probability with real-world predictions.
概率用0到1之间的尺度来衡量。理论概率假设所有结果是等可能的，而实验概率来自观测数据。样本空间图和系统列举能确保你数全所有结果。请记住互斥事件的加法法则和事件不发生的互补法则。通过练习求期望次数，你可以将概率与实际预测联系起来。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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