Mastering Probability for KS3: Basics, Experiments, and Rules | 掌握KS3概率：基础、实验与规则

📚 Mastering Probability for KS3: Basics, Experiments, and Rules | 掌握KS3概率：基础、实验与规则

Probability is the branch of mathematics that deals with how likely events are to happen. In KS3, you will learn to describe chance using numbers, words, and fractions, and to predict outcomes from simple experiments like flipping coins or rolling dice. This foundation prepares you for more complex statistics and helps you make sense of risk and uncertainty in everyday life, from weather forecasts to games of chance.

概率是数学中研究事件发生可能性的分支。在KS3阶段，你将学会用数字、词语和分数来描述机会，并通过掷硬币、掷骰子等简单实验预测结果。这一基础为你学习更复杂的统计学做好准备，并帮助你理解日常生活中的风险和不确定性，从天气预报到概率游戏。

1. What Is Probability? | 什么是概率？

Probability is a measure of how likely an event is to occur. It is always a number between 0 and 1, where 0 means the event is impossible and 1 means it is certain. For example, the probability of the sun rising tomorrow is 1 (certain), while the probability of flipping a coin and getting a “number 7” is 0 (impossible). Most events in real life have probabilities somewhere in between.

概率是衡量事件发生可能性的度量。它总是介于0和1之间的一个数，0表示事件不可能发生，1表示事件必然发生。例如，明天太阳升起的概率是1（必然），而掷一枚硬币得到“数字7”的概率是0（不可能）。现实生活中大多数事件的概率介于两者之间。

Probabilities can also be expressed as fractions, decimals, or percentages. The probability of getting heads when flipping a fair coin is ½, 0.5, or 50%. We use the notation P(event) to represent probability. For a fair six-sided die, P(rolling a 4) = &frac16;. Understanding this scale is the first step in working with chance.

概率也可以用分数、小数或百分比表示。抛掷一枚公平的硬币得到正面的概率是½、0.5或50%。我们用符号P(事件)来表示概率。对于一枚公平的六面骰子，P(掷出4) = &frac16;。理解这个尺度是处理机会问题的第一步。

2. The Probability Scale | 概率尺度

The probability scale is a number line from 0 to 1. Events that are impossible sit at 0, events that are certain sit at 1, and equally likely events (like getting a head or tail on a fair coin) sit at 0.5. Words like “unlikely”, “likely”, “even chance”, and “very unlikely” describe where probabilities fall on this scale. For instance, P(picking a red card from a standard deck) = ½, placing it at even chance.

概率尺度是一条从0到1的数轴。不可能事件位于0处，必然事件位于1处，等可能事件（如抛公平硬币得到正面或反面）位于0.5处。像“不太可能”“很可能”“机会均等”“极不可能”这样的词语描述了概率在尺度上的位置。例如，从标准扑克牌中抽到一张红牌的概率是½，处于机会均等的位置。

We can use the scale to compare probabilities. An event with probability 0.2 is less likely than one with probability 0.7. It is important to remember that a small probability does not mean an event will never happen – it just means it is unlikely in a single trial. The scale helps us visualise and order the likelihood of different outcomes.

我们可以用尺度来比较概率。概率为0.2的事件不如概率为0.7的事件容易发生。重要的是要记住，小概率并不意味着事件永远不会发生——它只意味着在单次试验中不太可能。尺度帮助我们直观地看到并对不同结果的可能性进行排序。

3. Basic Probability Formula | 基本概率公式

The probability of an event occurring can be calculated if we know all the possible outcomes and they are equally likely. The formula is:

P(Event) = Number of favourable outcomes ÷ Total number of possible outcomes

如果所有可能结果已知且等可能，我们可以计算事件发生的概率。公式为：

P(事件) = 有利结果的数量 ÷ 所有可能结果的总数

For example, when rolling a fair six-sided die, there are 6 possible outcomes. The favourable outcomes for rolling an even number are {2, 4, 6}, so there are 3 favourable outcomes. Thus, P(even) = 3 ÷ 6 = ½. This formula only works when every outcome is equally likely.

例如，掷一枚公平的六面骰子时，共有6种可能结果。掷出偶数的有利结果是{2, 4, 6}，所以有3个有利结果。因此，P(偶数) = 3 ÷ 6 = ½。该公式仅在所有结果等可能时才成立。

This formula is the foundation for calculating probabilities in simple experiments. Always check that outcomes are equally likely – for example, when drawing a card from a well-shuffled deck, each card has the same chance. The total number of outcomes is often called the ‘sample space’.

这个公式是计算简单实验概率的基础。一定要检查结果是否等可能——例如，从洗好的牌中抽一张牌，每张牌的机会都相同。可能结果的总数通常被称为“样本空间”。

4. Sample Space and Listing Outcomes | 样本空间与列出结果

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 die, it is {1, 2, 3, 4, 5, 6}. Listing outcomes systematically helps ensure we do not miss any. When two events happen together, like tossing two coins, we can use a sample space diagram or a table. The sample space for tossing two coins is {HH, HT, TH, TT}, where H = heads and T = tails. There are 4 equally likely outcomes.

样本空间是实验所有可能结果的集合。对于一次掷硬币，样本空间是{正面，反面}。对于掷骰子，样本空间是{1, 2, 3, 4, 5, 6}。系统地列出结果有助于确保不遗漏任何情况。当两个事件一起发生时，比如掷两枚硬币，我们可以使用样本空间图或表格。掷两枚硬币的样本空间是{HH, HT, TH, TT}，其中H表示正面，T表示反面。共有4种等可能的结果。

For more complex situations, we use two-way tables or tree diagrams. When rolling two dice, a 6×6 table shows all 36 outcomes. This method guarantees we count each combination exactly once. Being able to list outcomes clearly is a vital skill for finding probabilities of combined events.

对于更复杂的情况，我们使用双向表格或树形图。掷两枚骰子时，一个6×6的表格可以显示全部36种结果。这种方法确保我们每个组合只计一次。能够清晰地列出结果是求组合事件概率的关键技能。

5. Experimental Probability vs Theoretical Probability | 实验概率与理论概率

Theoretical probability is what we expect to happen based on equally likely outcomes. For a fair die, P(4) = &frac16;. Experimental probability is based on actual results from an experiment. If you roll a die 60 times and get a 4 on 12 of those rolls, the experimental probability of rolling a 4 is 12/60 = 0.2. These two values may differ because of chance variation in small samples.

理论概率是我们基于等可能结果预期发生的情况。对于公平的骰子，P(4) = &frac16;。实验概率基于实验的实际结果。如果你掷60次骰子，有12次得到4，那么掷出4的实验概率是12/60 = 0.2。这两个值可能因小样本中的随机波动而不同。

As the number of trials increases, the experimental probability tends to get closer to the theoretical probability. This is called the law of large numbers. In class, you might carry out experiments with coins or dice to see this in action. It reminds us that probability describes long-term behaviour, not short-term guarantees.

随着试验次数的增加，实验概率会趋向于理论概率。这被称为大数定律。在课堂上，你可以通过掷硬币或骰子的实验来观察这一现象。它提醒我们，概率描述的是长期行为，而非短期的保证。

6. Calculating Probabilities for Single Events | 单一事件概率的计算

We apply the basic formula to many single events. For a bag containing 3 red balls, 2 blue balls, and 5 green balls (total 10), the probability of picking a red ball is 3/10. The probability of picking a blue ball is 2/10 = 1/5. Remember to simplify fractions when possible. Probabilities should always be given in their simplest form unless a question specifies otherwise.

我们将基本公式应用于许多单一事件。对于一个装有3个红球、2个蓝球和5个绿球（共10个）的袋子，摸到红球的概率是3/10。摸到蓝球的概率是2/10 = 1/5。记得在可能时化简分数。除非题目另有要求，概率总是应以最简形式给出。

In a standard 52-card deck, P(drawing a King) = 4/52 = 1/13. P(drawing a Heart) = 13/52 = 1/4. Always identify the total number of outcomes and the number of ways the specific event can happen. This step-by-step approach reduces mistakes.

在一副标准的52张扑克牌中，P(抽到一张K) = 4/52 = 1/13。P(抽到一张红桃) = 13/52 = 1/4。始终要确定总结果数以及特定事件可能发生的方式数。这种分步骤的方法可以减少错误。

7. Probability of an Event NOT Happening | 事件不发生的概率

If the probability of an event happening is P(A), then the probability of it not happening is 1 – P(A). This is called the complement rule. The event “not A” is written as A’. For example, if the probability of rain tomorrow is 0.3, then the probability of no rain is 1 – 0.3 = 0.7.

如果事件发生的概率是P(A)，那么它不发生的概率就是1 – P(A)。这被称为互补规则。事件“非A”记作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。

This is very useful when calculating the complement is easier than calculating the event directly. For rolling a die, P(not a 6) = 1 – 1/6 = 5/6. The sum of the probabilities of all possible mutually exclusive outcomes is always 1. This principle helps us check our work.

当计算对立事件比直接计算事件更容易时，这非常有用。掷骰子时，P(不是6) = 1 – 1/6 = 5/6。所有互斥可能结果的概率之和总是1。这个原则帮助我们检查运算结果。

8. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法规则

Mutually exclusive events are events that cannot happen at the same time. For instance, when rolling a die, getting a 3 and getting a 5 are mutually exclusive. The probability of either event A or event B happening is P(A or B) = P(A) + P(B), provided they are mutually exclusive.

互斥事件是指不能同时发生的事件。例如，掷骰子时，得到3和得到5是互斥事件。事件A或事件B发生的概率是P(A或B) = P(A) + P(B)，前提是它们互斥。

In a bag of coloured marbles, P(red) = 0.2 and P(blue) = 0.3. These are mutually exclusive (you pick one marble). So, P(red or blue) = 0.2 + 0.3 = 0.5. If events are not mutually exclusive, we cannot simply add the probabilities—we must subtract the overlap, but that topic is usually covered later.

在一袋彩色弹珠中，P(红色) = 0.2，P(蓝色) = 0.3。这两个事件互斥（你只摸一颗弹珠）。所以，P(红色或蓝色) = 0.2 + 0.3 = 0.5。如果事件不是互斥的，我们不能简单相加——必须减去重叠部分，但那个主题通常会在以后学习。

9. Independent Events and the Multiplication Rule | 独立事件与乘法规则

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. The probability of both independent events A and B happening is P(A and B) = P(A) × P(B).

如果一个事件的结果不影响另一个事件的结果，则这两个事件是独立的。例如，掷硬币和掷骰子是独立事件。两个独立事件A和B都发生的概率是P(A和B) = P(A) × P(B)。

If P(Heads) = 0.5 and P(rolling a 6) = 1/6, then P(Heads and 6) = 0.5 × 1/6 = 1/12. This rule is key for solving problems involving multiple independent steps. Always check that the events really are independent—if they are not (like drawing cards without replacement), you must use conditional probability.

如果P(正面) = 0.5，而P(掷出6) = 1/6，那么P(正面且6) = 0.5 × 1/6 = 1/12。这个规则对于解决涉及多个独立步骤的问题至关重要。一定要检查事件是否真的独立——如果不独立（例如不放回地抽牌），你就必须使用条件概率。

10. Tree Diagrams for Combined Events | 组合事件的树形图

Tree diagrams help visualise sequences of events and calculate probabilities for combined outcomes. Each branch represents a possible outcome with its probability written along the branch. For two coin tosses, the first branch has two paths (H, T), each with probability 0.5. The second set of branches repeats this, giving four final outcomes with probability 0.25 each.

树形图有助于将一系列事件可视化，并计算组合结果的概率。每个分支代表一个可能的结果，其概率写在分支上。对于两次掷硬币，第一级分支有两条路径（H, T），每条概率为0.5。第二级分支重复这一过程，产生四种最终结果，每种概率为0.25。

To find the probability of a specific path, multiply the probabilities along the branches. For independent events, the tree diagram shows the multiplication rule in action. For example, the probability of getting two heads is 0.5 × 0.5 = 0.25. Tree diagrams are especially powerful when events are not independent, but in KS3 we mostly use them for independent events.

要找出某条路径的概率，将沿分支的概率相乘。对于独立事件，树形图展示了乘法规则的实际应用。例如，得到两个正面的概率是0.5 × 0.5 = 0.25。当事件不独立时，树形图尤其强大，但在KS3阶段我们主要将它们用于独立事件。

11. Relative Frequency and Probability Experiments | 相对频率与概率实验

Relative frequency is the experimental estimate of probability. It is calculated as (number of times the event occurred) ÷ (total number of trials). If you spin a spinner 50 times and it lands on blue 13 times, the relative frequency of blue is 13/50 = 0.26. This gives an estimated probability when theoretical probability is not known.

相对频率是概率的实验估计值。它的计算方法是（事件发生的次数）÷（试验总次数）。如果你旋转一个转盘50次，它停在蓝色区域13次，那么蓝色的相对频率是13/50 = 0.26。当理论概率未知时，这提供了一个估计概率。

In KS3, you will conduct experiments to compare theoretical and experimental probabilities. These activities demonstrate that with a larger number of trials, the relative frequency becomes a better estimator of the true probability. Recording results in a frequency table and plotting them on a graph helps visualise the convergence.

在KS3，你将进行实验来比较理论概率和实验概率。这些活动表明，随着试验次数的增加，相对频率会成为真实概率的更好估计值。将结果记录在频率表中并绘制图表有助于直观地看到收敛过程。

12. Expected Number of Outcomes | 期望结果数

If we know the probability of an event, we can predict how many times it should occur in a number of trials. The expected frequency is P(Event) × Number of trials. For example, if the probability of a biased coin showing heads is 0.4, and we toss it 200 times, we expect 0.4 × 200 = 80 heads.

如果我们知道一个事件的概率，就可以预测它在多次试验中应该发生的次数。期望频数是P(事件) × 试验次数。例如，如果一枚不均匀的硬币出现正面的概率是0.4，我们抛掷它200次，预期得到0.4 × 200 = 80次正面。

This does not guarantee exactly 80 heads; it is an average expectation over many repetitions. Expected value is a fundamental concept in probability and statistics, and it helps in making decisions under uncertainty. Use it to check whether experimental results are unusually high or low.

这并不保证恰好出现80次正面；它是多次重复试验下的平均期望值。期望值是概率与统计学中的一个基本概念，有助于在不确定性下做出决策。可以用它来检查实验结果是否异常偏高或偏低。

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Mastering Probability for KS3: Basics, Experiments, and Rules | 掌握KS3概率：基础、实验与规则