📚 Introduction to Probability | 概率入门
Probability is the branch of mathematics that deals with how likely events are to happen. It is used every day, from weather forecasts to games of chance, and it gives us a way to measure uncertainty. In KS3, you will learn to calculate probabilities as fractions, decimals, or percentages, and to use simple rules to work out the chance of combined events.
概率是数学中研究事件发生可能性大小的分支。从天气预报到运气游戏，概率每天都在被使用，它为我们提供了衡量不确定性的方法。在 KS3 阶段，你将学习如何用分数、小数或百分数计算概率，并运用简单规则计算复合事件发生的几率。
1. What Is Probability? | 什么是概率？
Probability is a number between 0 and 1 that indicates how likely an event is to occur. A probability of 0 means the event is impossible, while a probability of 1 means it is certain. Events that might happen but are not guaranteed have probabilities between 0 and 1.
概率是一个介于 0 和 1 之间的数，表示事件发生的可能性。概率为 0 表示事件不可能发生，概率为 1 表示事件必然发生。那些可能但不一定会发生的事件，其概率介于 0 和 1 之间。
For example, when you flip a fair coin, the probability of getting 'heads' is ½, or 0.5, or 50%. This means that if you flip the coin many times, you would expect heads to appear about half of the time.
例如，抛一枚均匀硬币时，得到“正面”的概率是 ½，即 0.5 或 50%。这意味着如果你抛掷硬币很多次，你预计正面会出现大约一半的次数。
2. The Probability Scale | 概率标度
We can place the probability of any event on a scale from 0 to 1. Events such as 'the sun will rise tomorrow' are almost certain and have a probability very close to 1. Events like 'picking a red ball from a bag of only blue balls' are impossible and have probability 0.
我们可以把任何事件的概率放在从 0 到 1 的标尺上。“明天太阳会升起”这样的事件几乎是完全确定的，概率非常接近 1。而“从一个只有蓝球的袋子里摸出红球”这样的事件则是不可能的，概率为 0。
Words like 'likely', 'unlikely', 'even chance', and 'certain' can be mapped onto the probability scale. An even chance corresponds to a probability of 0.5. An event with probability 0.2 is unlikely, and an event with probability 0.8 is likely.
像“很可能”、“不太可能”、“公平的机会”和“确定”这样的词语可以对应到概率标尺上。公平的机会对应概率 0.5。概率为 0.2 的事件不太可能发生，概率为 0.8 的事件则很可能发生。
3. Basic Probability Formula | 基本概率公式
The probability of an event happening can be found using the formula:
事件发生的概率可以用公式求出：
Probability = Number of favourable outcomes ÷ Total number of possible outcomes
This works as long as all outcomes are equally likely. For example, when rolling a fair six‑sided die, each face has an equal chance of landing face up, so the probability of rolling a 4 is 1/6.
这个公式在所有结果发生的可能性相等时才有效。例如，掷一枚均匀的六面骰子时，每一面朝上的机会相等，所以掷出 4 的概率是 1/6。
Always check that the outcomes are equally likely. If you use a biased die, this simple formula does not apply because some faces are more likely than others.
一定要检查结果是否等可能发生。如果使用一枚偏心骰子，简单的公式就不适用了，因为有些面比其他面更容易出现。
4. Expressing Probability | 概率的表示
Probabilities can be written as fractions, decimals, or percentages. The fraction form is often preferred because it shows exact values. For example, the probability of drawing a heart from a standard deck of 52 playing cards is 13/52, which simplifies to ¼.
概率可以用分数、小数或百分数来表示。分数形式往往更受青睐，因为它能显示精确值。例如，从一副标准的 52 张扑克牌中抽到红桃的概率是 13/52，化简后为 ¼。
To convert a probability fraction to a decimal, divide the numerator by the denominator. So ¼ becomes 0.25. To express this as a percentage, multiply by 100 to get 25%. All these forms are equally valid.
要将概率分数转换为小数，用分子除以分母。¼ 就成为 0.25。要表达为百分数，乘以 100 得到 25%。所有这些形式都同样有效。
5. Complementary Events | 互补事件
the probability of rain tomorrow is 0.3, then the probability of no rain is 1 − 0.3 = 0.7.
一个事件的补集是该事件不发生的情况。一个事件与其互补事件的概率之和总是 1。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。
This rule is very useful when it is easier to calculate the probability that something does NOT happen. For example, to find the probability of getting at least one head in two coin flips, it is easier to first find the probability of getting no heads (two tails) and subtract from 1.
当计算某事件不发生的概率更容易时，这条规则就非常有用。例如，要计算抛两枚硬币至少得到一次正面的概率，更容易的方法是先求出没有正面的概率（两次都是反面），再从 1 中减去。
6. Sample Space Diagrams | 样本空间图
6 × 6 table showing all 36 outcomes helps to count the favourable outcomes for events like 'sum of 7'.
样本空间是一个实验所有可能结果的列表。组织样本空间的一种方法是使用表格或系统的清单。对于掷两个骰子，一个 6 × 6 的表格显示所有 36 种结果，有助于计算像“和为 7”这样的事件的有利结果数。
Another type of diagram is the possibility space diagram, which is essentially the same as a sample space table. It allows you to visualise the equally likely outcomes and avoid missing any.
另一种图示是可能空间图，本质上与样本空间表相同。它能让你直观看到等可能发生的结果，避免遗漏任何可能。
7. Experimental Probability vs Theoretical Probability | 实验概率与理论概率
Theoretical probability is what we expect to happen based on the number of equally likely outcomes. Experimental probability (or relative frequency) is found by actually carrying out an experiment or survey.
理论概率是基于等可能结果的数量，我们预期将要发生的情况。实验概率（或相对频率）是通过实际进行实验或调查得出的。
Experimental probability = Number of times the event occurs ÷ Total number of trials
.167. As the number of trials increases, experimental probability often gets closer to the theoretical probability.
如果你掷骰子 100 次，出现 6 的次数是 22 次，那么掷出 6 的实验概率是 22/100 = 0.22。理论概率是 1/6 ≈ 0.167。随着试验次数增加，实验概率通常会趋近于理论概率。
8. 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 odd number' and 'roll an even number' are mutually exclusive because a single roll cannot be both odd and even.
如果两个事件不能同时发生，它们就是互斥事件。例如，掷一枚骰子时，“掷出奇数”和“掷出偶数”是互斥的，因为一次掷骰不可能既是奇数又是偶数。
P(A or B) = P(A) + P(B). This is known as the addition rule for mutually exclusive events.
对于互斥事件，事件 A 或事件 B 发生的概率是它们各自概率之和：P(A 或 B) = P(A) + P(B)。这被称为互斥事件的加法法则。
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 because the coin flip does not change what number appears on the die.
如果一个事件的结果不会影响另一个事件的结果，那么这两个事件就是独立的。例如，抛硬币和掷骰子是独立的，因为抛硬币不会改变骰子出现的点数。
d B) = P(A) × P(B). So the probability of getting heads and rolling a 3 is ½ × 1/6 = 1/12.
对于独立事件，A 和 B 同时发生的概率是它们各自概率的乘积：P(A 且 B) = P(A) × P(B)。因此，得到正面且掷出 3 的概率是 ½ × 1/6 = 1/12。
10. Using Tree Diagrams | 使用树形图
A tree diagram is a helpful tool for showing all the outcomes of an experiment that has two or more stages. Each branch represents a possible outcome, and we write the probability along the branch. To find the probability of a combined outcome, we multiply along the branches.
树形图是展示具有两个或多个阶段实验的所有结果的有效工具。每条树枝代表一个可能的结果，我们把概率写在树枝旁边。要找出组合结果的概率，我们将沿途的概率相乘。
Tail, each probability ½) and second flip (Head or Tail, each probability ½). The outcome 'Head then Tail' has probability ½ × ½ = ¼.
例如，抛两次硬币的树形图有两组分支：第一次抛（正面或反面，概率均为 ½）和第二次抛（正面或反面，概率均为 ½）。“先正面再反面”的结果概率为 ½ × ½ = ¼。
Tree diagrams also help when events are not independent, but at KS3, we mainly deal with independent events.
当事件不是独立时，树形图也有帮助，但在 KS3 阶段，我们主要处理独立事件。
11. Expected Frequency | 期望频数
Expected frequency is the number of times we would expect an event to happen if we repeated an experiment a large number of times. It is calculated by multiplying the probability of the event by the number of trials.
期望频数是如果我们大量重复一项实验，预计事件会发生的次数。它由事件发生的概率乘以试验次数计算得出。
Expected frequency = Probability × Number of trials
200 = 80 times. Note that the actual result may be different because of chance variation.
如果一枚偏心硬币落地正面的概率是 0.4，你抛掷它 200 次，你预期正面出现 0.4 × 200 = 80 次。请注意，由于偶然性，实际结果可能会有所不同。
12. Probability in Real Life | 生活中的概率
Probability is used in many real‑life situations, such as calculating insurance premiums, testing new medicines, predicting weather, and designing fair games. Understanding the basics of probability helps you to assess risks and make informed decisions.
概率被用于许多现实生活情景，比如计算保险费、测试新药、预测天气和设计公平的游戏。理解概率的基础知识有助于你评估风险，做出明智的决定。
When you see a statement like 'there is a 70% chance of rain', it is based on probability models that analyse past data and current conditions. Learning probability gives you the tools to interpret such information critically.
当你看到“有 70% 的可能下雨”这样的陈述时，它是基于分析过去数据和当前状况的概率模型得出的。学习概率将赋予你批判性地解读这类信息的工具。
Published by TutorHao | Probability Revision Series | aleveler.com
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