📚 Probability Scale and Single Events | 概率标度与单一事件
Probability is the branch of mathematics that measures how likely an event is to happen. In KS3 Cambridge Mathematics, students learn to describe chance using numbers, fractions, decimals and percentages, and to calculate simple probabilities from equally likely outcomes. This article covers the probability scale, key vocabulary, the probability formula, complementary events, sample spaces and expected frequency, all with worked examples and practice ideas.
概率是数学中衡量某个事件发生可能性大小的分支。在 KS3 剑桥数学中，学生要学会用数字、分数、小数和百分比描述机会，并根据等可能结果计算简单概率。本文涵盖概率标度、关键术语、概率公式、互补事件、样本空间与期望频率，并配有详细示例和练习思路。
1. The Probability Scale | 概率标度
Probability is always written as a number between 0 and 1. A probability of 0 means an event is impossible, while a probability of 1 means an event is certain. Values close to 0 describe unlikely events, and values close to 1 describe likely events. The middle value 0.5, or 1/2, represents an even chance.
概率总是写成 0 到 1 之间的一个数。概率为 0 表示事件不可能发生，概率为 1 表示事件一定发生。接近 0 的值描述不太可能发生的事件，接近 1 的值描述很可能发生的事件。中间值 0.5，即 1/2，表示机会均等。
Probabilities can be expressed as fractions, decimals or percentages. For example, a probability of 0.25 is the same as 1/4 or 25%. You may be asked to convert between these forms, so you should be confident with equivalent fractions, decimals and common percentages.
概率可以用分数、小数或百分比表示。例如，0.25 的概率等于 1/4 或 25%。考试中可能要求在这些形式之间转换，因此需要熟练等价分数、小数和常见百分比。
- Impossible: P = 0
- Certain: P = 1
- Even chance: P = 0.5 = 1/2 = 50%
- 不可能：P = 0
- 必然发生：P = 1
- 机会均等：P = 0.5 = 1/2 = 50%
0 ≤ P(event) ≤ 1
2. Key Vocabulary: Outcomes, Events and Sample Space | 关键术语：结果、事件与样本空间
An outcome is one possible result of a probability experiment, such as rolling a 3 on a fair dice. An event is a set of one or more outcomes, such as rolling an odd number. The sample space is the list of all possible outcomes in an experiment.
结果是一次概率实验可能出现的一个结果，例如掷一个均匀骰子得到 3。事件是一个或多个结果的集合，例如掷出奇数。样本空间是实验中所有可能结果的列表。
For a fair six-sided dice, the sample space is {1, 2, 3, 4, 5, 6}. If the event is ‘rolling an even number’, the favourable outcomes are 2, 4 and 6. Clear vocabulary helps you set up probability questions correctly.
对于一个均匀的六面骰子，样本空间是 {1, 2, 3, 4, 5, 6}。如果事件是“掷出偶数”，有利结果是 2、4 和 6。清晰的术语有助于正确建立概率问题。
3. The Probability Formula | 概率计算公式
When all outcomes are equally likely, probability is calculated using a simple formula. You count the number of favourable outcomes and divide it by the total number of possible outcomes.
当所有结果等可能时，概率可以用一个简单公式计算。数出有利结果的个数，再除以所有可能结果的总数。
P(event) = number of favourable outcomes ÷ total number of outcomes
Example: A bag contains 3 red balls, 2 blue balls and 5 green balls. The total number of balls is 10. The probability of picking a red ball at random is 3/10, because there are 3 favourable outcomes out of 10 total outcomes.
示例：一个袋子里有 3 个红球、2 个蓝球和 5 个绿球。球的总数是 10。随机取出一个红球的概率是 3/10，因为 10 个总结果中有 3 个有利结果。
Always simplify fractions when possible. If the probability of an event is 4/8, it should normally be written as 1/2 or 0.5. Examiners often expect probabilities in their simplest form.
分数应尽量化简。如果某个事件的概率是 4/8，通常应写成 1/2 或 0.5。考官一般要求概率用最简形式表示。
4. Probability as a Fraction, Decimal and Percentage | 概率的分数、小数与百分比形式
The same probability can be written in three forms. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100.
同一个概率可以写成三种形式。将分数转换为小数，用分子除以分母。将小数转换为百分比，乘以 100。
| Fraction | Decimal | Percentage | Meaning |
|---|---|---|---|
| 1/4 | 0.25 | 25% | Unlikely |
| 1/2 | 0.5 | 50% | Even chance |
| 3/4 | 0.75 | 75% | Likely |
Being able to move between these forms is useful when comparing probabilities. For example, 0.3, 35% and 2/5 should be compared by converting them all to decimals: 0.3, 0.35 and 0.4.
能够在这些形式之间转换，对比较概率很有帮助。例如，比较 0.3、35% 和 2/5，可以都转换为小数：0.3、0.35 和 0.4。
5. Complementary Events and the Probability of ‘Not A’ | 互补事件与“非 A”的概率
The complement of an event A is the event that A does not happen. It is written as A’ or ‘not A’. The probabilities of an event and its complement always add up to 1.
事件 A 的补集是 A 不发生的事件，写作 A’ 或“非 A”。一个事件与其补集的概率之和总是等于 1。
P(not A) = 1 − P(A)
Example: The probability that it rains tomorrow is 0.3. The probability that it does not rain is 1 − 0.3 = 0.7. This is a very common KS3 question type because it often provides a quicker route than listing all unfavourable outcomes.
示例：明天下雨的概率是 0.3。不下雨的概率是 1 − 0.3 = 0.7。这是 KS3 常见题型，因为它通常比列出所有不利结果更快捷。
6. Fair Spinners and Dice | 均匀转盘与骰子
A fair spinner or dice has equally likely outcomes. This means each number or sector has the same chance of being selected. When a question says ‘fair’, the probability formula can be used directly.
均匀转盘或骰子具有等可能结果。这意味着每个数字或扇区被选中的机会相同。当题目出现“均匀”时，可以直接使用概率公式。
Example: A fair eight-sided spinner has numbers 1 to 8. The probability of spinning a multiple of 3 is 2/8 = 1/4, because the multiples of 3 are 3 and 6. The probability of spinning a number less than 5 is 4/8 = 1/2, because the favourable outcomes are 1, 2, 3 and 4.
示例：一个均匀的八面转盘标有数字 1 到 8。转到 3 的倍数的概率是 2/8 = 1/4，因为 3 的倍数是 3 和 6。转到小于 5 的数的概率是 4/8 = 1/2，因为有利结果是 1、2、3 和 4。
7. Listing Outcomes and Sample Space Diagrams | 列出结果与样本空间图
For two-part experiments, such as flipping two coins or rolling two dice, it is helpful to list all outcomes in a sample space diagram. This prevents missing or double-counting outcomes.
对于两步实验，例如抛两枚硬币或掷两个骰子，列出样本空间图中的所有结果会很有帮助。这样可以防止遗漏或重复计算结果。
Example: Flipping two coins gives the sample space {HH, HT, TH, TT}. The probability of getting at least one head is 3/4, because three of the four outcomes contain an H. The probability of getting two tails is 1/4.
示例：抛两枚硬币得到的样本空间为 {HH, HT, TH, TT}。至少得到一次正面的概率是 3/4，因为四个结果中有三个包含 H。得到两个反面的概率是 1/4。
For rolling two dice, a 6 by 6 table can show all 36 outcomes. This is useful for questions such as finding the probability that the total is 7 or that both dice show the same number.
掷两个骰子时，可以用 6×6 表格展示所有 36 个结果。这对于求总数为 7 或两个骰子数字相同的概率等问题很有用。
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
8. 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, rolling a dice and getting an odd number and getting an even number are mutually exclusive. The probability of either one happening is found by adding their individual probabilities.
如果两个事件不能同时发生，则它们互斥。例如，掷一个骰子得到奇数和得到偶数是互斥的。任一事件发生的概率等于各自概率之和。
P(A or B) = P(A) + P(B) for mutually exclusive A and B
Example: A bag contains 2 red, 3 blue and 5 green marbles. P(red) = 2/10, P(blue) = 3/10. Since red and blue are mutually exclusive, P(red or blue) = 2/10 + 3/10 = 5/10 = 1/2.
示例：一个袋子里有 2 个红球、3 个蓝球和 5 个绿球。P(红) = 2/10，P(蓝) = 3/10。由于红球和蓝球互斥，P(红或蓝) = 2/10 + 3/10 = 5/10 = 1/2。
Be careful: this simple addition rule only works when events are mutually exclusive. If events can overlap, the overlap must be subtracted, which is covered later at IGCSE level.
注意：只有当事件互斥时，这个简单加法法则才适用。如果事件可能重叠，则必须减去重叠部分，这一内容将在 IGCSE 阶段学习。
9. Expected Frequency | 期望频率
Probability can predict how many times an event is likely to occur in a repeated experiment. The expected frequency is found by multiplying the probability of the event by the number of trials.
概率可以预测在重复实验中某个事件可能发生的次数。期望频率等于事件概率乘以试验次数。
Expected frequency = P(event) × number of trials
Example: The probability of spinning a 2 on a fair spinner is 1/5. If the spinner is spun 200 times, the expected number of 2s is 1/5 × 200 = 40. This does not guarantee exactly 40 twos, but it is the long-run average.
示例：在一个均匀转盘上转到 2 的概率是 1/5。如果转盘旋转 200 次，转到 2 的期望次数是 1/5 × 200 = 40。这并不能保证恰好出现 40 次 2，但这是长期平均值。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
A common mistake is writing a probability greater than 1 or less than 0. Remember that all probabilities must lie between 0 and 1 inclusive. If your answer is 1.2 or −0.3, check your working immediately.
常见错误是写出大于 1 或小于 0 的概率。记住所有概率必须介于 0 和 1 之间（含端点）。如果答案是 1.2 或 −0.3，要立即检查计算过程。
Another mistake is forgetting to simplify fractions or using the wrong denominator. Always count the total outcomes carefully and check that all outcomes are equally likely. If a dice is not fair, you cannot simply use 1/6 for each face.
另一个错误是忘记化简分数或使用错误的分母。务必仔细数出总结果数，并检查所有结果是否等可能。如果骰子不均匀，就不能简单地将每个面都视为 1/6。
- Check that 0 ≤ probability ≤ 1
- Simplify fractions where possible
- Identify sample space before calculating
- Use complementary events for ‘not’ questions
- Show clear working and label P(A)
- 检查 0 ≤ 概率 ≤ 1
- 尽可能化简分数
- 计算前先确定样本空间
- “非”类问题使用互补事件
- 写出清晰步骤并标注 P(A)
11. Worked Practice Questions | 典型练习题
Question 1: A letter is chosen at random from the word ‘MATHEMATICS’. Find the probability that it is the letter M.
问题 1：从单词 “MATHEMATICS” 中随机选取一个字母。求选到字母 M 的概率。
There are 11 letters in total. The letter M appears twice. Therefore P(M) = 2/11. This fraction cannot be simplified further.
总共有 11 个字母。字母 M 出现两次。因此 P(M) = 2/11。这个分数不能再化简。
Question 2: A fair 20-sided dice numbered 1 to 20 is rolled once. Find the probability that the score is a prime number greater than 5.
问题 2：一个标有数字 1 到 20 的均匀二十面骰子掷一次。求掷出的数是一个大于 5 的质数的概率。
The prime numbers greater than 5 and up to 20 are 7, 11, 13, 17 and 19. There are 5 favourable outcomes. The total number of outcomes is 20, so P = 5/20 = 1/4.
大于 5 且不超过 20 的质数是 7、11、13、17 和 19。有 5 个有利结果。总结果数是 20，所以 P = 5/20 = 1/4。
Question 3: The probability of a train being late is 0.12. Find the probability that the train is not late.
问题 3：火车晚点的概率是 0.12。求火车不晚点的概率。
P(not late) = 1 − 0.12 = 0.88. This can also be written as 88% or 22/25.
P(不晚点) = 1 − 0.12 = 0.88。这也可以写成 88% 或 22/25。
12. Summary | 小结
Probability measures chance on a scale from 0 to 1. Use the formula P(event) = favourable outcomes ÷ total outcomes when all outcomes are equally likely. The complement rule P(not A) = 1 − P(A) is a powerful shortcut. Sample space diagrams help with listing outcomes for two-step experiments. Expected frequency links probability with repeated trials. Master these core ideas and you will have a strong foundation for IGCSE probability.
概率是在 0 到 1 的标度上衡量机会的大小。当所有结果等可能时，使用公式 P(事件) = 有利结果数 ÷ 总结果数。互补法则 P(非 A) = 1 − P(A) 是一个强大的捷径。样本空间图有助于列出两步实验的结果。期望频率将概率与重复试验联系起来。掌握这些核心概念，你将为 IGCSE 概率打下坚实基础。
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