Probability: Single Events and Experimental Probability | 概率：单事件与实验概率

📚 Probability: Single Events and Experimental Probability | 概率：单事件与实验概率

Welcome to this in-depth revision guide on probability, inspired by the type of exercises found in typical KS3 Cambridge mathematics worksheets such as p233_1.pdf. Probability helps us measure how likely an event is to happen, from flipping coins to rolling dice and beyond. In this article, we will cover single events, experimental probability, and essential rules that form the foundation for more advanced topics.

欢迎阅读这份深入的概率复习指南，灵感来源于典型 KS3 剑桥数学练习题（例如 p233_1.pdf）。概率帮助我们衡量事件发生的可能性，从抛硬币到掷骰子等等。在本文中，我们将涵盖单事件、实验概率以及构成更高级主题基础的重要规则。

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1. The Meaning of Probability | 概率的含义

Probability is a measure of the chance that a particular event will occur. It is expressed as a number between 0 and 1, where 0 means an impossible event and 1 means a certain event. In KS3 mathematics, we often write probabilities as fractions, decimals, or percentages.

概率是对特定事件发生可能性的度量。它用一个介于 0 到 1 之间的数字表示，其中 0 表示不可能事件，1 表示必然事件。在 KS3 数学中，我们通常将概率写成分数、小数或百分数。

For example, the probability of the Sun rising tomorrow is essentially 1, while the probability of rolling a 7 on a standard 6-sided dice is 0.

例如，太阳明天升起的概率本质上是 1，而在标准六面骰子上掷出 7 的概率是 0。

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2. The Probability Scale | 概率尺度

Probabilities can be placed on a scale from 0 to 1. A probability of ½ or 0.5 indicates an even chance, like getting heads when you toss a fair coin. Marking events on a probability line helps you compare likelihoods. For example, the probability of rolling a 6 on a fair dice is ⅙, which is closer to 0 than to 1, so it is unlikely.

概率可以放在从 0 到 1 的尺度上。概率为 ½ 或 0.5 表示机会均等，例如抛一枚公平硬币得到正面。在概率线上标记事件有助于比较可能性。例如，在一个公平的骰子上掷出 6 的概率是 ⅙，更接近 0 而非 1，因此是不太可能发生的。

Words such as impossible, unlikely, even chance, likely, and certain can be mapped to numbers: 0, ¼, ½, ¾, and 1, respectively.

不可能、不太可能、均等、很可能和必然等词语可对应到数字：0、¼、½、¾ 和 1。

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3. Theoretical Probability Formula | 理论概率公式

When all outcomes are equally likely, the theoretical probability of an event A is given by:

当所有结果等可能时，事件 A 的理论概率由下式给出：

P(A) = Number of favourable outcomes / Total number of possible outcomes

P(A) = 有利结果的数量 / 所有可能结果的总数

This simple fraction forms the backbone of KS3 probability work. Always simplify the fraction where possible. For example, if you pick a card from a standard 52-card deck, the probability of getting a king is ⁴/₅₂ = ¹/₁₃.

这个简单的分数构成了 KS3 概率学习的支柱。请尽可能简化分数。例如，从一副标准的 52 张扑克牌中抽一张牌，抽到 K 的概率是 ⁴/₅₂ = ¹/₁₃。

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4. Sample Spaces and Outcomes | 样本空间与结果

The sample space is the set of all possible outcomes of an experiment. For a single dice roll, the sample space is {1, 2, 3, 4, 5, 6}. For flipping two coins, it is {HH, HT, TH, TT}. Organising the sample space systematically, often using a list or a table, ensures you count all outcomes correctly.

样本空间是实验所有可能结果的集合。掷一个骰子的样本空间是 {1, 2, 3, 4, 5, 6}。抛两枚硬币的样本空间是 {HH, HT, TH, TT}。系统地组织样本空间（通常使用列表或表格）可确保正确计数所有结果。

Understanding the sample space is crucial for calculating probabilities of combined events, such as the sum of two dice.

理解样本空间对于计算组合事件（例如两个骰子点数之和）的概率至关重要。

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5. Fair Dice and Equal Likelihood | 公平骰子与等可能性

A fair dice is one where each face has the same chance of landing face up. For a standard 6-sided dice, the probability of any specific number, say 3, is ⅙. Because the outcomes are equally likely, the theoretical formula works perfectly.

公平骰子指的是每个面朝上的机会均等。对于标准的六面骰子，掷出任意特定数字（比如 3）的概率是 ⅙。由于结果等可能，理论公式完美适用。

If the dice is biased, some numbers appear more often, and theoretical probability will not match experimental results. We then rely on experimental probability, which you will learn about next.

如果骰子有偏，某些数字出现得更频繁，理论概率将不会与实验结果相匹配。此时我们依赖实验概率，接下来你将学习这一点。

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6. Complementary Events | 互补事件

The complement of an event A is the event ‘not A’, denoted A’. For any event, P(A) + P(A’) = 1. So if the probability of rain tomorrow is 0.3, the probability it does not rain is 0.7. This rule is extremely useful for solving ‘at least one’ type problems.

事件 A 的互补事件是 “非 A”，记作 A’。对于任何事件，P(A) + P(A’) = 1。因此，如果明天下雨的概率是 0.3，那么不下雨的概率就是 0.7。这一规则在解决“至少一次”类型问题时非常有用。

For example, the probability of getting at least one head in two coin flips is 1 – P(all tails) = 1 – ¼ = ¾.

例如，抛两枚硬币至少得到一个正面的概率是 1 – P(全是反面) = 1 – ¼ = ¾。

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7. Experimental Probability and Relative Frequency | 实验概率与相对频率

Experimental probability is based on actual trials or experiments. If you roll a dice 60 times and get a 4 on 12 occasions, the relative frequency of a 4 is 12/60 = ⅕. As the number of trials increases, the experimental probability tends to get closer to the theoretical probability – this is known as the law of large numbers.

实验概率基于实际试验或实验。如果你掷一个骰子 60 次，有 12 次得到 4，那么 4 的相对频率是 12/60 = ⅕。随着试验次数增加，实验概率往往接近理论概率——这被称为大数定律。

In KS3 investigations, you often compare theoretical and experimental probabilities by conducting experiments and recording tallies.

在 KS3 探究活动中，你经常通过进行实验并记录计数来比较理论概率与实验概率。

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8. Expected Frequency | 期望频数

If you know the probability of an event and the number of trials, you can calculate the expected frequency: Expected frequency = probability × number of trials. For example, if you toss a fair coin 200 times, you would expect heads about ½ × 200 = 100 times.

如果你知道一个事件的概率和试验次数，就可以计算期望频数：期望频数 = 概率 × 试验次数。例如，如果你抛一枚公平硬币 200 次，你期望大约 ½ × 200 = 100 次正面。

This does not guarantee exactly 100 heads, but it gives an average over many repetitions.

这并不保证正好 100 次正面，但给出了多次重复的平均值。

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9. Probability Tree Diagrams for Single Events | 单事件概率树图

Tree diagrams help visualise sequences of events. Starting with a single event, branches show possible outcomes and their probabilities. For two independent events, you multiply probabilities along branches. A simple tree for tossing a coin twice has two branches for the first toss (H, T) and then another set for the second toss.

树图有助于将事件序列可视化。从单事件开始，各分支显示可能的结果及其概率。对于两个独立事件，沿着分支相乘概率。一个抛两次硬币的简单树图在第一次抛掷时有两条分支（H、T），然后在第二次抛掷时又分出一组。

At KS3, you mainly draw trees for events like picking coloured counters from a bag with replacement. This introduces the multiplication rule.

在 KS3 阶段，你主要会为类似于从袋子里有放回地抽取彩色筹码这样的事件画树图。这引入了乘法规则。

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10. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法法则

Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a dice, getting a 2 and getting a 5 are mutually exclusive. If events A and B are mutually exclusive, then P(A or B) = P(A) + P(B). So the probability of rolling a 2 or a 5 is ⅙ + ⅙ = ²/₆ = ⅓.

如果两个事件不能同时发生，则它们是互斥的。例如，掷骰子时，得到 2 和得到 5 是互斥的。如果事件 A 和 B 互斥，那么 P(A 或 B) = P(A) + P(B)。因此掷出 2 或 5 的概率是 ⅙ + ⅙ = ²/₆ = ⅓。

If events are not mutually exclusive, you must subtract the overlap: P(A or B) = P(A) + P(B) – P(A and B). This is touched upon more in later KS3 topics.

如果事件不是互斥的，则必须减去重叠部分：P(A 或 B) = P(A) + P(B) – P(A 且 B)。这在后续 KS3 主题中会有所涉及。

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11. Independent Events and the Multiplication Rule | 独立事件与乘法法则

Independent events are those where the outcome of one does not affect the outcome of the other. For independent events A and B, P(A and B) = P(A) × P(B). For instance, the probability of flipping a coin and getting heads, and rolling a dice and getting a 6, is ½ × ⅙ = ¹/₁₂.

独立事件指的是一个事件的结果不影响另一个事件的结果。对于独立事件 A 和 B，P(A 且 B) = P(A) × P(B)。例如，抛硬币得到正面且掷骰子得到 6 的概率是 ½ × ⅙ = ¹/₁₂。

Watch out for events that are not independent, such as picking two cards from a deck without replacement; then the probabilities change after the first pick.

注意非独立事件，例如从一副牌中无放回地抽取两张牌；此时在第一次抽取后概率会发生变化。

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12. Common Mistakes and Tips | 常见错误与提示

Students often forget that probabilities must be between 0 and 1, or they add probabilities incorrectly for non-mutually exclusive events. Another common pitfall is confusing ‘and’ with ‘or’: ‘and’ requires multiplication, ‘or’ often requires addition. Always check if outcomes are equally likely before applying the theoretical formula.

学生常忘记概率值必须在 0 到 1 之间，或者对非互斥事件错误地进行概率相加。另一个常见陷阱是混淆“且”与“或”：“且”需要用乘法，“或”通常用加法。在应用理论公式之前，请务必检查结果是否等可能。

Practice by drawing sample space diagrams or lists. And remember, experimental probability is only an estimate; it becomes more reliable with more trials.

通过绘制样本空间图或列表来练习。记住，实验概率只是一个估计值；试验次数越多，它就越可靠。

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