📚 Probability and Sample Space | 概率与样本空间
Probability is a way of measuring how likely something is to happen. It helps us make predictions when the outcome is uncertain. In this article, you will learn to describe chance using numbers, words and fractions, draw sample space diagrams, and use probability rules to solve real-life problems. Mastering these skills will prepare you for more advanced topics in statistics and decision-making.
概率是衡量某件事发生可能性大小的一种方式。当结果不确定时，它能帮助我们对未来做出预测。在这篇文章中，你将学习用数字、词语和分数来描述机会，绘制样本空间图，并运用概率法则解决实际生活中的问题。掌握这些技能将为你学习更高级的统计与决策内容做好准备。
1. What is Probability? | 什么是概率？
Probability is the branch of mathematics that deals with chance and uncertainty. Whenever you toss a coin, roll a dice or spin a spinner, the outcome is not known in advance, but you can still say something about how likely each possible result is. Probability gives you a way to measure that likelihood on a scale from 0 to 1.
概率是研究机会和不确定性的数学分支。每当你抛硬币、掷骰子或转动转盘时，结果事先并不知道，但你仍然可以描述每种可能结果出现的可能性有多大。概率提供了一种在0到1之间衡量这种可能性的方法。
2. The Probability Scale | 概率尺度
The probability scale runs from 0 (impossible) to 1 (certain). An event that is just as likely to happen as not has a probability of ½ or 0.5. Probabilities can be written as fractions, decimals or percentages. The table below shows some common descriptions and their probability values.
概率尺度从0（不可能）到1（必然发生）。一件发生与不发生的可能性相等的事件，概率为½或0.5。概率可以用分数、小数或百分数表示。下面的表格展示了一些常见的描述及其概率值。
| Description | Probability (fraction) | Probability (decimal) |
|---|---|---|
| Impossible | 0 | 0 |
| Unlikely | ¼ | 0.25 |
| Even chance | ½ | 0.5 |
| Likely | ¾ | 0.75 |
| Certain | 1 | 1 |
3. Expressing Probability | 概率的表示
Probabilities can be shown in words (certain, likely, even chance, unlikely, impossible), as a fraction, as a decimal or as a percentage. For example, when you roll a fair six-sided dice, the probability of rolling a 3 is 1 out of 6, which is written as P(3) = 1/6, about 0.167 or 16.7%.
概率可以用词语（一定、很可能、对等机会、不太可能、不可能）来表示，也可以用分数、小数或百分数表示。例如，当你掷一个均匀的六面骰子时，掷出3点的概率是6份中的1份，记作 P(3) = 1/6，约等于0.167或16.7%。
4. Calculating Simple Probabilities | 计算简单概率
When all outcomes are equally likely, the probability of an event is found using the formula:
P(Event) = Number of favourable outcomes ÷ Total number of outcomes
当所有结果发生的可能性都相等时，事件的概率可以用以下公式求出：
P(事件) = 有利结果的数量 ÷ 所有结果的总数
e probability of picking a blue sweet at random is 2 ÷ (3+2+5) = 2/10 = 1/5 or 0.2.
例如，一个袋子里装有3颗红色糖果、2颗蓝色糖果和5颗绿色糖果。随机取出一颗蓝色糖果的概率是 2 ÷ (3+2+5) = 2/10 = 1/5 或 0.2。
5. Sample Space Diagrams | 样本空间图
A sample space is the set of all possible outcomes of an experiment. You can list the sample space systematically or use a table (sample space diagram) to show all combinations when two events happen together, such as rolling two dice or tossing a coin and spinning a spinner.
样本空间是一次试验中所有可能结果的集合。你可以系统地列出样本空间，或者用表格（样本空间图）来表示两个事件同时发生时的所有组合，比如掷两个骰子，或者抛一枚硬币并转动一个转盘。
Here is a sample space diagram for rolling two fair dice and adding the scores:
下面是掷两个均匀骰子并将点数相加的样本空间图：
| + | 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 |
ally likely pairs, you can see that the probability of getting a total of 7 is 6/36 = 1/6.
从36组等可能的结果中可以看出，总点数为7的概率是 6/36 = 1/6。
6. Probability from Two-Way Tables | 从双向表求概率
Two-way tables help organise information about two categories at once. For example, a survey of 100 students shows how many prefer football or basketball and whether they are in Year 7 or Year 8. You can then find probabilities like P(Year 7 student who prefers football) by reading directly from the table.
双向表可以同时呈现两种分类的信息。例如，一项关于100名学生的调查显示了他们喜欢足球还是篮球，以及他们是在七年级还是八年级。然后你就可以直接从表格中读出像P(喜欢足球的七年级学生)这样的概率。
7. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. When you roll a dice, getting a 2 and getting a 5 are mutually exclusive – you cannot roll both with one throw. For mutually exclusive events, the probability of one event or the other occurring is the sum of their individual probabilities:
P(A or B) = P(A) + P(B)
如果两个事件不可能同时发生，它们就是互斥的。掷骰子时，掷出2点和掷出5点就是互斥事件——你不可能一次掷出两个点数。对于互斥事件，其中一个事件或另一个事件发生的概率等于它们各自概率之和：
P(A 或 B) = P(A) + P(B)
If P(2) = 1/6 and P(5) = 1/6, then P(2 or 5) = 1/6 + 1/6 = 1/3.
如果 P(2) = 1/6，P(5) = 1/6，那么 P(2或5) = 1/6 + 1/6 = 1/3。
8. Complementary Events | 互补事件
Two outcomes are complementary if one of them must happen, and they cover all possibilities without overlap. The probability of an event happening plus the probability of it not happening always equals 1. This is written as:
P(Event) + P(not Event) = 1
如果两个结果中必定有一个发生，并且它们覆盖了所有可能性且没有重叠，那么它们就是互补的。一个事件发生的概率与它不发生的概率之和总是等于1。这可以写成：
P(事件) + P(非事件) = 1
it will rain tomorrow is 0.3, then the probability that it will not rain is 1 - 0.3 = 0.7.
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。
9. Experimental Probability | 实验概率
Experimental probability is based on actual trials or experiments rather than theory. It is calculated as:
erimental probability = Number of times the event occurred ÷ Total number of trials
实验概率基于实际的试验或实验，而不是理论。它的计算公式为：
实验概率 = 事件发生的次数 ÷ 试验总次数
e theoretical probability of 0.5, but with more trials the experimental probability usually gets closer to the theoretical value.
例如，如果你抛一枚硬币50次，得到28次正面，那么正面的实验概率就是 28/50 = 0.56。这可能与0.5的理论概率不同，但随着试验次数增加，实验概率通常会趋近于理论值。
10. Expected Frequency | 期望频数
Once you know the theoretical probability of an event, you can predict how many times it should happen over many trials. This is called the expected frequency, found by multiplying the probability by the number of trials:
Expected frequency = P(Event) × Number of trials
一旦知道事件的理论概率，就可以预测它在多次试验中应该发生的次数。这就是期望频数，用概率乘以试验总次数求得：
期望频数 = P(事件) × 试验总次数
l a fair dice 600 times, the expected frequency of rolling a 4 is (1/6) × 600 = 100.
如果你掷一个均匀的骰子600次，掷出4点的期望频数是 (1/6) × 600 = 100。
11. Using Probability in Real Life | 在实际生活中应用概率
Probability is used everywhere – weather forecasting, insurance, games, medical testing and even in making everyday decisions. Understanding probability helps you assess risks and make informed choices. For example, knowing the chance of rain helps you decide whether to carry an umbrella, and understanding the odds in a game helps you play strategically.
概率无处不在——天气预报、保险、游戏、医学检测，甚至在日常生活决策中都会用到。理解概率可以帮助你评估风险并做出明智的选择。例如，知道下雨的可能性可以帮你决定是否需要带伞，而理解游戏中的胜率则能让你更有策略地参与。
12. Summary | 总结
Probability measures how likely events are on a scale from 0 to 1. Key ideas include the probability formula for equally likely outcomes, sample space diagrams for two events, mutually exclusive and complementary events, and the link between experimental and theoretical probability. Practice using these tools with tables, lists and simple experiments, and you will build a strong foundation for later work in statistics and data analysis.
概率用0到1的尺度衡量事件发生的可能性。关键知识点包括等可能结果下的概率公式、两个事件的样本空间图、互斥事件和互补事件，以及实验概率与理论概率之间的联系。通过表格、列表和简单实验来练习使用这些工具，你将为今后的统计与数据分析学习打下坚实的基础。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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