📚 Introduction to Probability | 概率入门
Probability is the branch of mathematics that describes how likely an event is to happen. Whether you are rolling a die, picking a card from a deck, or predicting tomorrow’s weather, probability gives you a way to measure uncertainty. In KS3, you will learn to express probability as a fraction, decimal, or percentage, and you will explore the difference between experimental and theoretical probability. Understanding these ideas is not only essential for exams but also helps you make better decisions in everyday life, from games of chance to risk assessment.
概率是数学中描述事件发生可能性大小的分支。无论你是掷骰子、从一副牌中抽牌，还是预测明天的天气，概率都为你提供了一种衡量不确定性的方法。在 KS3 阶段，你将学习用分数、小数或百分数来表示概率，并探索实验概率与理论概率的区别。理解这些概念不仅对考试至关重要，还能帮助你在日常生活中做出更好的决策，从机会游戏到风险评估。
1. What is Probability? | 什么是概率？
Probability is a number between 0 and 1 that tells us how likely an event is to occur. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain. For example, the probability that the sun will rise tomorrow is essentially 1, and the probability of rolling a 7 on a standard six-sided die is 0. In mathematics, we often write the probability of an event A as P(A).
概率是一个介于 0 和 1 之间的数，它告诉我们一个事件发生的可能性有多大。概率为 0 意味着事件不可能发生，概率为 1 意味着事件必然发生。例如，太阳明天升起的概率基本上是 1，而在一颗标准六面骰子上掷出 7 点的概率是 0。在数学中，我们通常将事件 A 的概率写作 P(A)。
2. The Probability Scale | 概率尺度
You can picture probability on a number line from 0 to 1. Words like ‘impossible’, ‘unlikely’, ‘even chance’, ‘likely’, and ‘certain’ can be placed along this scale. For instance, flipping a fair coin and getting heads has an even chance (0.5), while rolling a number greater than 1 on a die is likely. Learning to match probabilities to these descriptions helps build intuition.
你可以把概率想象成一条从 0 到 1 的数轴。诸如“不可能”“不太可能”“机会均等”“有可能”“必然”等词语可以放置在这条尺度上。例如，抛一枚公平的硬币得到正面是机会均等（0.5），而掷骰子得到大于 1 的点数是很有可能发生的。学会将概率与这些描述匹配有助于培养直觉。
3. Expressing Probability | 表示概率
Probability can be written as a fraction, a decimal, or a percentage. For example, the probability of getting a head on a fair coin can be written as ½, 0.5, or 50%. In many problems, fractions are preferred because they show exact values without rounding. The formula for theoretical probability is:
Probability = Number of favourable outcomes ÷ Total number of possible outcomes
概率可以用分数、小数或百分数表示。例如，抛一枚公平的硬币得到正面的概率可以写为 ½、0.5 或 50%。在许多问题中，分数更受欢迎，因为它们显示精确值而无需四舍五入。理论概率的公式是：
概率 = 有利结果的数量 ÷ 所有可能结果的总数
4. Sample Space | 样本空间
The sample space is the set of all possible outcomes of an experiment. For a single coin toss, the sample space is {Heads, Tails}. For rolling a fair six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. Listing outcomes systematically helps you avoid missing any possibilities, especially when dealing with combined events such as rolling two dice.
样本空间是一次实验所有可能结果的集合。对于单次抛硬币，样本空间是 {正面，反面}。对于掷一个公平的六面骰子，样本空间是 {1, 2, 3, 4, 5, 6}。系统地列出结果有助于避免遗漏任何可能性，尤其是在处理两个骰子这样的组合事件时。
5. Equally Likely Outcomes | 等可能结果
Many probability problems assume that all outcomes in the sample space are equally likely. This means that each outcome has the same chance of occurring. A fair die, for instance, gives each number a probability of ⅙. If outcomes are not equally likely, you cannot simply count favourable outcomes over total outcomes; you need a different approach, often using experiments or given data.
许多概率问题假设样本空间中的所有结果都是等可能的。这意味着每个结果发生的概率相同。例如，一颗公平的骰子给出每个点数的概率是 ⅙。如果结果不是等可能的，你就不能简单地用有利结果除以总结果来计数；你需要不同的方法，通常利用实验或给定的数据。
6. Theoretical vs. Experimental Probability | 理论概率与实验概率
Theoretical probability is what should happen based on the structure of the experiment, assuming all outcomes are equally likely. Experimental probability comes from actually performing the experiment and recording results. For example, the theoretical probability of rolling a 3 on a die is ⅙, but if you roll a die 60 times and get a 3 twelve times, the experimental probability is 12/60 = ⅕. As the number of trials increases, experimental probability tends to get closer to theoretical probability, a principle known as the law of large numbers.
理论概率是根据实验的结构、假设所有结果等可能时应该发生的情况。实验概率来自实际进行实验并记录结果。例如，掷骰子得到 3 点的理论概率是 ⅙，但如果你掷骰子 60 次并得到 3 点 12 次，实验概率就是 12/60 = ⅕。随着试验次数的增加，实验概率会趋向于接近理论概率，这一原理被称为大数定律。
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, the events ‘roll an even number’ and ‘roll an odd number’ are mutually exclusive. The probability of one or the other of two mutually exclusive events happening is the sum of their individual probabilities: P(A or B) = P(A) + P(B). This is called the addition rule for mutually exclusive events.
如果两个事件不可能同时发生，则它们是互斥的。例如，掷骰子时，“掷出偶数”和“掷出奇数”这两个事件就是互斥的。发生两个互斥事件中的一个的概率是它们各自概率的和：P(A 或 B) = P(A) + P(B)。这被称为互斥事件的加法法则。
8. The Sum of All Probabilities | 所有概率之和
In any sample space, the probabilities of all possible outcomes must add up to 1. This can be used to find missing probabilities. For example, if a biased coin has a 0.6 chance of landing heads, then the chance of landing tails must be 1 − 0.6 = 0.4, because heads and tails are the only outcomes and they are mutually exclusive. Similarly, if you have a spinner with colours red, blue, and green, and P(red)=½, P(blue)=⅓, then P(green)=1 − ½ − ⅓ = ⅙.
在任何样本空间中，所有可能结果的概率之和必须等于 1。这可以用来求出缺失的概率。例如，如果一枚偏斜硬币有 0.6 的概率正面着地，那么反面着地的概率一定是 1 − 0.6 = 0.4，因为正面和反面是仅有的结果且它们是互斥的。同样，如果你有一个转盘，颜色有红、蓝、绿，且 P(红)=½，P(蓝)=⅓，那么 P(绿)=1 − ½ − ⅓ = ⅙。
9. Independent Events | 独立事件
Two events are independent if the outcome of one does not affect the outcome of the other. For example, flipping a coin and rolling a die are independent events. To find the probability of two independent events both happening, you multiply their probabilities: P(A and B) = P(A) × P(B). So the probability of getting heads and rolling a 5 is ½ × ⅙ = 1/12. Careful: do not confuse independence with mutual exclusivity—they are different concepts.
如果两个事件中一个的结果不影响另一个的结果，则称它们是独立的。例如，抛硬币和掷骰子是独立事件。要求两个独立事件同时发生的概率，你将它们的概率相乘：P(A 与 B) = P(A) × P(B)。因此，得到正面并掷出 5 点的概率是 ½ × ⅙ = 1/12。注意：不要把独立性与互斥性混淆——它们是不同的概念。
10. Using Tree Diagrams | 使用树形图
Tree diagrams are a powerful visual tool for showing all possible outcomes of two or more events and their probabilities. Each branch represents a possible outcome, and you multiply along the branches to find the probability of a sequence. For example, a bag contains 3 red and 2 blue marbles; if you pick two marbles with replacement, the probability of picking red then blue is (3/5) × (2/5) = 6/25. Tree diagrams help you organise complex situations, especially when events are not independent.
树形图是一种强大的可视化工具，用于显示两个或多个事件的所有可能结果及其概率。每个分支代表一个可能的结果，你沿着分支相乘得到一系列事件的概率。例如，一个袋子里有 3 颗红色和 2 颗蓝色弹珠；如果你有放回地抽取两次，那么先红后蓝的概率是 (3/5) × (2/5) = 6/25。树形图帮助你理清复杂的情形，尤其是当事件不是独立的时候。
11. Expected Number of Successes | 预期的成功次数
If you know the probability of an event, you can predict how many times it should happen in a given number of trials. The expected number is simply: probability × number of trials. For instance, if the probability of rain on any day is 0.3, then in 100 days you would expect rain on 0.3 × 100 = 30 days. This is a handy way to check whether experimental results are close to what theory predicts.
如果你知道一个事件的概率，你可以预测在给定次数的试验中它应该发生多少次。预期的次数就是：概率 × 试验次数。例如，如果任何一天下雨的概率是 0.3，那么在 100 天中，你预期会下雨的天数是 0.3 × 100 = 30 天。这是一个检验实验结果是否接近理论预测的简便方法。
12. Common Misconceptions in Probability | 概率中的常见误区
One common mistake is the gambler’s fallacy: believing that past outcomes affect future independent events, like thinking a coin is ‘due’ to land heads after several tails. Another is assuming all outcomes are equally likely when they are not, such as treating a biased die as fair. Also, adding probabilities for non-mutually exclusive events without adjusting for overlap leads to double counting. Always check whether events are independent, mutually exclusive, and equally likely before applying rules.
一个常见的错误是赌徒谬误：认为过去的结果会影响未来的独立事件，比如觉得一枚硬币在连续几次反面后“该”出正面了。另一个误区是当结果并非等可能时却假设它们是等可能的，例如将一颗偏斜的骰子当作公平的来处理。还有，对于非互斥事件，不调整重叠部分就直接相加概率会导致重复计算。在套用规则之前，一定要检查事件是否独立、互斥以及等可能。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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