📚 Probability and Venn Diagrams | 概率与维恩图
Probability is the branch of mathematics that measures how likely an event is to happen. It is used every day in weather forecasts, games of chance, medical tests and decision making. In the Cambridge KS3 Mathematics curriculum, you need to understand the probability scale, calculate simple probabilities, use sample space diagrams, and represent combined events with Venn diagrams and two-way tables.
概率是数学中用来衡量某个事件发生可能性大小的一个分支。它广泛应用于天气预报、机会游戏、医学检测和决策制定中。在剑桥 KS3 数学课程中，你需要理解概率标度、计算简单概率、使用样本空间图，并能用维恩图和双向表表示复合事件。
1. What is Probability? | 什么是概率？
Probability gives a numerical value to the chance that a particular event will occur. It is always expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.
概率用数值表示某个特定事件发生的可能性大小。它始终用 0 到 1 之间的数来表示，其中 0 表示事件不可能发生，1 表示事件一定会发生。
For example, the probability of getting a head when tossing a fair coin is 1/2 because there are two equally likely outcomes: heads and tails. The probability of rolling a number less than 7 on a normal six-sided dice is 1 because all six outcomes are less than 7.
例如，抛一枚公平硬币得到正面的概率是 1/2，因为只有正面和反面两种等可能结果。掷一枚普通六面骰子得到小于 7 的数字的概率是 1，因为所有六个结果都小于 7。
2. The Probability Scale | 概率标度
The probability scale is a number line from 0 to 1. Events can be placed on this scale to show how likely they are. A probability close to 0 means very unlikely, while a probability close to 1 means very likely.
概率标度是一条从 0 到 1 的数轴。可以把事件放在这条标度上表示它们的可能性大小。概率接近 0 表示非常不可能，概率接近 1 表示非常可能。
- Impossible event: probability = 0
- Certain event: probability = 1
- Equally likely event: probability = 0.5
- 不可能事件：概率 = 0
- 必然事件：概率 = 1
- 等可能事件：概率 = 0.5
ilities can be written as fractions, decimals or percentages. For example, 1/4 = 0.25 = 25%. In KS3 you should be comfortable converting between these three forms.
概率可以写成分数、小数或百分数。例如，1/4 = 0.25 = 25%。在 KS3 阶段，你应该能够熟练地在三种形式之间进行转换。
3. The Basic Probability Formula | 基本概率公式
When all outcomes in a sample space are equally likely, the probability of an event A is calculated using the formula:
当样本空间中的所有结果等可能时，事件 A 的概率用以下公式计算：
P(A) = n(A) ÷ n(S)
Here n(A) represents the number of outcomes in event A, and n(S) represents the total number of outcomes in the sample space S. The symbol P(A) means 'the probability of A'.
其中 n(A) 表示事件 A 中结果的个数，n(S) 表示样本空间 S 中结果的总数。符号 P(A) 表示“事件 A 的概率”。
For 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 is 3/10, because there are 3 red balls and 10 balls in total.
例如，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。摸到红球的概率是 3/10，因为有 3 个红球，总共有 10 个球。
4. Sample Space Diagrams | 样本空间图
A sample space is the set of all possible outcomes of an experiment. A sample space diagram is a useful way to list all outcomes clearly, especially when two events happen at the same time, such as rolling two dice or flipping two coins.
样本空间是指一次试验中所有可能结果的集合。样本空间图是一种清晰地列出所有结果的方法，尤其是当两个事件同时发生时，例如掷两个骰子或抛两枚硬币。
For two coins, the sample space can be written as {HH, HT, TH, TT}, where H means heads and T means tails. Each outcome has a probability of 1/4 if the coins are fair.
对于两枚硬币，样本空间可以写成 {HH, HT, TH, TT}，其中 H 表示正面，T 表示反面。如果硬币是公平的，则每个结果的概率都是 1/4。
6 because there are six ways to make a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
掷两个骰子时，可以用 6×6 的表格表示所有 36 种可能的点数组合或有序列对。得到和为 7 的概率是 6/36 = 1/6，因为组成和为 7 的六种方式为：(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)。
5. 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 dice, the events 'rolling a 2' and 'rolling a 5' are mutually exclusive because a single roll cannot show both numbers at once.
如果两个事件不能同时发生，则称它们为互斥事件。例如，掷一枚骰子时，“掷出 2”和“掷出 5”是互斥事件，因为一次掷骰不可能同时出现两个数字。
For mutually exclusive events A and B, the probability that A or B occurs is the sum of their individual probabilities:
对于互斥事件 A 和 B，事件 A 或 B 发生的概率等于它们各自概率之和：
P(A or B) = P(A) + P(B)
green counters. If P(red) = 0.3 and P(blue) = 0.2, then P(red or blue) = 0.3 + 0.2 = 0.5.
例如，一个袋子里只有红色、蓝色和绿色三种计数器。如果 P(红) = 0.3，P(蓝) = 0.2，则 P(红或蓝) = 0.3 + 0.2 = 0.5。
6. Independent Events | 独立事件
Two events are independent if the outcome of one event does not affect the outcome of the other. For example, rolling a dice and flipping a coin are independent events because the dice result does not change the coin result.
如果两个事件中一个事件的结果不会影响另一个事件的结果，则这两个事件是独立事件。例如，掷骰子和抛硬币是独立事件，因为骰子的结果不会改变硬币的结果。
For independent events A and B, the probability that both A and B occur is found by multiplying their individual probabilities:
对于独立事件 A 和 B，事件 A 和 B 同时发生的概率等于它们各自概率的乘积：
P(A and B) = P(A) × P(B)
two independent draws occur, the probability of winning in both draws is 0.1 × 0.1 = 0.01.
例如，在一次抽奖中赢得小奖的概率是 0.1。如果进行两次独立抽奖，则两次都中奖的概率是 0.1 × 0.1 = 0.01。
7. Venn Diagrams | 维恩图
A Venn diagram uses overlapping circles inside a rectangle to show relationships between sets. The rectangle represents the universal set, which includes all possible outcomes. Each circle represents a specific event or set of outcomes.
维恩图用矩形内重叠的圆来表示集合之间的关系。矩形表示全集，包含所有可能的结果。每个圆代表一个特定事件或一组结果。
For example, in a class of 30 students, 12 play football and 9 play basketball. If 4 students play both sports, a Venn diagram can show how many students play only football, only basketball, both sports, or neither sport.
例如，在一个有 30 名学生的班级里，12 人踢足球，9 人打篮球。如果 4 名学生两项运动都参加，则维恩图可以显示有多少学生只踢足球、只打篮球、两项都参加或两项都不参加。
Numbers placed in each region of a Venn diagram must add up to the total number in the universal set. This helps solve problems involving 'and', 'or' and 'neither'.
维恩图每个区域中的数字相加必须等于全集中的总数。这有助于解决涉及“且”、“或”和“既不……也不……”的问题。
8. Intersection and Union | 交集与并集
The intersection of two sets A and B, written A ∩ B, contains all outcomes that belong to both A and B. In a Venn diagram, this is the overlapping region of the two circles.
两个集合 A 和 B 的交集写作 A ∩ B，包含所有既属于 A 又属于 B 的结果。在维恩图中，这是两个圆重叠的区域。
The union of two sets A and B, written A ∪ B, contains all outcomes that belong to A or B or both. In a Venn diagram, this is any region covered by either circle.
两个集合 A 和 B 的并集写作 A ∪ B，包含所有属于 A 或 B 或同时属于两者的结果。在维恩图中，这是被任意一个圆覆盖的所有区域。
The probability of A or B for non-mutually exclusive events is given by:
对于非互斥事件，A 或 B 的概率由下式给出：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
This formula prevents double counting the outcomes that lie in the intersection. You should use it when the two events can occur at the same time.
这个公式可以避免对交集部分的结果重复计数。当两个事件可以同时发生时，你应该使用这个公式。
9. Complement of an Event | 事件的补集
The complement of an event A, written A' or Aᶜ, is the event that A does not happen. It includes every outcome in the sample space that is not in A.
事件 A 的补集写作 A' 或 Aᶜ，表示事件 A 不发生的情况。它包含样本空间中所有不属于 A 的结果。
The probability of the complement is calculated by subtracting P(A) from 1:
补集的概率通过用 1 减去 P(A) 来计算：
P(A') = 1 − P(A)
ins tomorrow is 0.3, then the probability that it does not rain tomorrow is 1 − 0.3 = 0.7.
例如，如果明天下雨的概率是 0.3，那么明天不下雨的概率就是 1 − 0.3 = 0.7。
Using the complement is often quicker when an event has many possible outcomes. Instead of adding many probabilities, you can subtract the unwanted probability from 1.
当事件有很多可能结果时，使用补集通常更快捷。你可以用 1 减去不需要的概率，而不用把许多概率加起来。
10. Two-Way Tables | 双向表
A two-way table organizes data by two categories and helps calculate probabilities involving 'and' and 'or'. Each row and column total gives useful information about the whole group.
双向表按两个类别整理数据，有助于计算涉及“且”和“或”的概率。每一行和每一列的总和提供了有关整组数据的有用信息。
| Category | Boys | Girls | Total |
|---|---|---|---|
| Own a pet | 12 | 15 | 27 |
| No pet | 8 | 5 | 13 |
| Total | 20 | 20 | 40 |
ho owns a pet is 12/40 = 3/10. The probability that a student does not own a pet is 13/40.
从这张表中可以看出，随机选择一名养宠物的男生的概率是 12/40 = 3/10。学生不养宠物的概率是 13/40。
Always read two-way table questions carefully. Decide whether you are choosing from the whole group or from a particular row or column before calculating the probability.
做双向表题目时一定要仔细审题。在计算概率之前，要先确定你是从整个群体中选取，还是从某一行或某一列中选取。
11. Common Misconceptions | 常见误区
Many KS3 students add probabilities when they should multiply, or multiply when they should add. Remember: 'or' usually means add, while 'and' usually means multiply, but only when the events are mutually exclusive or independent respectively.
许多 KS3 学生在该相乘时用了加法，在该相加时用了乘法。请记住：“或”通常表示相加，“且”通常表示相乘，但前提是事件分别为互斥事件或独立事件。
Another common mistake is forgetting that probabilities from all possible outcomes must add to 1. If your probabilities sum to more than 1, you have made an error.
另一个常见错误是忘记所有可能结果的概率之和必须等于 1。如果你的概率之和大于 1，就说明计算出错了。
Students sometimes count outcomes in a sample space incorrectly, especially with two dice, because they forget that (1,2) and (2,1) are different outcomes unless the problem says otherwise.
学生在计算样本空间结果时有时会出错，尤其是在掷两个骰子的情况下，因为他们忘记了 (1,2) 和 (2,1) 是不同的结果，除非题目另有说明。
12. Exam-Style Tips and Summary | 考试技巧与总结
In Cambridge KS3 checkpoint and progression tests, probability questions often combine fractions, decimals and percentages. Show all working clearly, simplify fractions where possible, and write final answers in the form requested by the question.
在剑桥 KS3 checkpoint 和进阶测试中，概率题常常结合分数、小数和百分数。要清楚地写出所有解题过程，尽可能化简分数，并按照题目要求的形式写出最终答案。
When using a Venn diagram, label each region with its value before answering. Check that all the numbers add up to the universal total. This simple check will catch many careless errors.
使用维恩图时，先在每个区域标上数值再作答。检查所有数字加起来是否等于全集总数。这个简单的检查可以发现很多粗心错误。
- Probability measures chance from 0 to 1.
- Sample space diagrams list all outcomes.
- Mutually exclusive events use P(A or B) = P(A) + P(B).
- Independent events use P(A and B) = P(A) × P(B).
- Venn diagrams show intersections, unions and complements.
- Always check that probabilities add to 1.
- 概率衡量从 0 到 1 的可能性大小。
- 样本空间图列出所有可能结果。
- 互斥事件使用 P(A 或 B) = P(A) + P(B)。
- 独立事件使用 P(A 且 B) = P(A) × P(B)。
- 维恩图表示交集、并集和补集。
- 始终检查概率之和是否等于 1。
By mastering these core ideas, you will be able to tackle a wide range of KS3 probability questions with confidence and accuracy.
掌握这些核心概念后，你将能够自信、准确地解决各种 KS3 概率问题。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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