📚 KS3 Cambridge Mathematics: Probability | 剑桥KS3数学：概率
Probability is the branch of mathematics that describes how likely an event is to occur. In KS3 Cambridge Mathematics, you learn to calculate probabilities, predict outcomes, and represent chance using fractions, decimals, and percentages. This guide covers all essential concepts, from the probability scale to tree diagrams, helping you master the topic for Cambridge Checkpoint and beyond.
概率是描述事件发生可能性的数学分支。在剑桥 KS3 数学中，你将学习如何计算概率、预测结果，并使用分数、小数和百分比表示可能性。本指南涵盖了从概率标度到树状图的所有基本概念，帮助你在剑桥 Checkpoint 考试及更高阶段掌握这一主题。
1. What is Probability? | 什么是概率？
Probability measures the chance that a particular event will happen. It is always a number between 0 and 1 inclusive. An event with probability 0 is impossible, while an event with probability 1 is certain to happen. In real life, we rarely deal with absolute certainty, so most probabilities lie somewhere in between. For example, the probability of the sun rising tomorrow is very close to 1, but we might treat it as certain in most calculations.
概率衡量某一特定事件发生的可能性。它始终是一个介于 0 和 1 之间的数字。概率为 0 的事件是不可能发生的，而概率为 1 的事件是必然发生的。在现实生活中，我们很少遇到绝对确定的情况，因此大多数概率值介于两者之间。例如，明天太阳升起的概率非常接近 1，但在多数计算中我们可将其视为必然事件。
The concept of probability is fundamental to statistics and helps us make decisions under uncertainty. When you toss a coin, the outcome appears random, yet probability gives us a reliable way to describe the long-term behaviour of such random experiments. This blend of randomness and predictability is what makes probability so powerful in mathematics and science.
概率的概念是统计学的基础，有助于我们在不确定性下做决策。抛硬币时，结果看似随机，但概率为我们提供了一种可靠的方法来描述此类随机实验的长期行为。这种随机与可预测的结合，正是概率在数学和科学中如此强大的原因。
2. Probability Scale | 概率标度
The probability scale places events on a line from 0 to 1. We can label the scale with words such as impossible, unlikely, even chance, likely, and certain. For example, tossing a fair coin and getting heads has an even chance (probability = 1/2 or 0.5). Rolling a standard die and getting a number less than 7 is certain (probability = 1). Landing on a section of a spinner with no colour would be impossible (probability = 0).
概率标度将事件放置在从 0 到 1 的一条直线上。我们可以用文字标注标度，如不可能、不太可能、均等机会、很可能和必然。例如，抛一枚公平硬币得到正面有均等机会（概率 = 1/2 或 0.5）。掷一个标准骰子得到的数字小于 7 是必然事件（概率 = 1）。转盘指针停在无色区域是不可能事件（概率 = 0）。
In Cambridge assessments, you may be asked to mark probabilities on a scale or interpret a given scale. Remember to use fractions, decimals, or percentages interchangeably. For instance, a probability of 0.75 can also be written as 3/4 or 75%. Being comfortable with all three representations is an important skill because exam questions may ask you to express a probability in a specified format. Always double-check that your answer lies between 0 and 1, otherwise an error has been made.
在剑桥评估中，你可能需要在标度上标记概率或解释给定的标度。记得可以互换使用分数、小数或百分比。例如，0.75 的概率也可以写作 3/4 或 75%。熟练运用这三种表示形式是一项重要技能，因为考题可能要求你用指定格式表达概率。务必检查答案是否在 0 和 1 之间，否则就出错了。
3. Calculating Basic Probability | 计算基本概率
To find the probability of an event, use the formula:
计算事件概率的公式：
P(Event) = Number of favourable outcomes / Total number of possible outcomes
P(事件) = 有利结果的数量 / 所有可能结果的总数
For example, when you roll a fair six-sided die, the probability of rolling a 3 is 1/6 because there is one favourable outcome (3) and six possible outcomes (1, 2, 3, 4, 5, 6). The probability of rolling an even number is 3/6 = 1/2, since the even numbers are 2, 4, and 6. Always simplify fractions where possible, and express the result in its simplest form. The probability fraction must be between 0 and 1.
例如，掷一枚公平的六面骰子，掷出 3 的概率是 1/6，因为有利结果（3）只有一个，而可能结果共有六个（1、2、3、4、5、6）。掷出偶数的概率是 3/6 = 1/2，因为偶数为 2、4、6。尽可能将分数化简，用最简形式表示结果。概率分数必须在 0 和 1 之间。
A related concept is the complement rule: the probability of an event not happening is 1 minus the probability that it does happen. Symbolically, P(not A) = 1 − P(A). For instance, if the probability of rain tomorrow is 0.3, then the probability it will not rain is 1 − 0.3 = 0.7. This simple rule is extremely useful when calculating the probability of the complementary event is easier than calculating the event itself.
一个相关概念是互补规则：事件 不 发生的概率，等于 1 减去该事件发生的概率。用符号表示为 P(非 A) = 1 − P(A)。例如，若明天下雨的概率是 0.3，则不下雨的概率是 1 − 0.3 = 0.7。当计算互补事件的概率比直接计算该事件更容易时，这个简单规则极为有用。
4. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot occur at the same time. For example, when rolling a die, the events ‘rolling a 1’ and ‘rolling a 6’ are mutually exclusive because you cannot roll both numbers at once. The probability of either event happening is the sum of their individual probabilities: P(A or B) = P(A) + P(B).
如果两个事件不能同时发生，则称它们是互斥的。例如，掷骰子时，“掷出 1”和“掷出 6”这两个事件是互斥的，因为你不可能同时掷出两个数字。任一事件发生的概率是它们各自概率之和：P(A 或 B) = P(A) + P(B)。
If events are not mutually exclusive, you must subtract the overlap to avoid double counting, but at KS3 level, most problems involve mutually exclusive events. A typical question might ask: ‘A spinner has four equal sections labelled 2, 3, 5 and 8. What is the probability of getting an even number or a prime number?’ Since the outcomes 2 (even and prime) and 3,5 (prime) and 8 (even) occur, and 2 belongs to both sets, you must use the general addition rule. However, if the events are defined carefully as ‘even number’ and ‘odd prime’, they become mutually exclusive and the simple sum works.
如果事件不是互斥的，则必须减去重叠部分以避免重复计算，但在 KS3 阶段，大多数问题涉及互斥事件。一道典型的题目可能这样问：“一个转盘有四个相等的区域，分别标有数字 2、3、5 和 8。转到偶数或质数的概率是多少？”由于结果 2（既是偶数又是质数）以及 3、5（质数）和 8（偶数）都会出现，且 2 同属两个集合，因此你需要使用一般的加法法则。然而，若将事件谨慎定义为“偶数”和“奇质数”，它们便成为互斥事件，简单的求和运算就可行了。
5. Experimental Probability | 实验概率
Experimental probability (also called relative frequency) is based on actual trials or experiments. It is calculated as:
实验概率（也称相对频率）基于实际的试验或实验。其计算方式为：
Experimental Probability = Number of times the event occurred / Total number of trials
实验概率 = 事件发生的次数 / 试验总次数
For instance, if you flip a coin 50 times and get heads 22 times, the experimental probability of heads is 22/50 = 11/25. This may differ from the theoretical probability of 1/2. As the number of trials increases, experimental probability usually gets closer to the theoretical probability – a concept known as the law of large numbers. In class experiments, you often compare your experimental results with theoretical predictions and discuss reasons for any discrepancy, such as bias or insufficient trials.
例如，抛硬币 50 次得到正面 22 次，则正面的实验概率为 22/50 = 11/25。这可能与理论概率 1/2 不同。随着试验次数增加，实验概率通常会趋近于理论概率——这一概念称为大数定律。在课堂实验中，你经常需要比较实验结果与理论预测，并讨论出现偏差的原因，比如硬币的偏倚或试验次数不足等。
6. Expected Frequency | 期望频次
Expected frequency predicts how many times an event is likely to happen in a set number of trials. The formula is:
期望频次预测在一组试验中事件可能发生的次数。公式为：
Expected Frequency = Probability of the event × Number of trials
期望频次 = 事件发生的概率 × 试验次数
Suppose the probability that a light bulb is defective is 0.02. If a factory produces 1000 bulbs, the expected number of defective bulbs is 0.02 × 1000 = 20. Expected frequency does not guarantee the exact outcome but gives an average over many repetitions. This concept helps in quality control, genetics predictions, and game strategies. Always check whether you are given a theoretical probability or an experimental one; the same formula applies, but you must use the correct probability value for the context.
假设一个灯泡有缺陷的概率是 0.02。如果工厂生产 1000 个灯泡，预计有缺陷的灯泡数为 0.02 × 1000 = 20。期望频次不保证
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