📚 Probability: Sample Spaces and Simple Events | 概率: 样本空间与简单事件
Probability helps us answer questions about chance and uncertainty in everyday life, from flipping a coin to predicting the weather. In this article, you will learn what probability means, how to describe and list all possible outcomes for simple experiments, and how to calculate the chance of an event using a straightforward formula.
概率帮助我们回答日常生活中关于机会和不确定性的问题，从抛硬币到预测天气。在本文中，你将学习概率的含义，如何描述和列出简单实验的所有可能结果，以及如何使用一个简单的公式计算某个事件发生的可能性。
1. What is Probability? | 概率是什么？
Probability is a number that describes how likely an event is to happen. It always lies between 0 and 1. A probability of 0 means the event cannot happen at all, and a probability of 1 means the event is absolutely certain to happen. Most probabilities lie somewhere in between, and they can be written as fractions, decimals or percentages.
概率是描述一个事件发生可能性的数字。它的值总是在0和1之间。概率为0意味着事件绝不可能发生，概率为1意味着事件必然发生。大多数概率值介于两者之间，并且可以用分数、小数或百分数表示。
2. Simple Events and Outcomes | 简单事件与结果
An outcome is a single possible result of an experiment. For example, when you roll an ordinary six‑sided die, the outcomes are 1, 2, 3, 4, 5 and 6. A simple event is a collection of one or more outcomes. Rolling an even number is an event that includes the outcomes 2, 4 and 6.
结果是实验的一个可能结果。例如，当你掷一个普通的六面骰子时，可能的结果是1、2、3、4、5和6。简单事件是由一个或多个结果组成的集合。掷出偶数是包含了结果2、4和6的一个事件。
3. Sample Space | 样本空间
The set of all possible outcomes for an experiment is called the sample space. We often use the letter S or the Greek letter Ω (omega) to denote it. For rolling a fair die once, the sample space is S = {1, 2, 3, 4, 5, 6}. Knowing the sample space is the first step to calculating any probability because it tells us the total number of equally likely outcomes.
一个实验所有可能结果的集合被称为样本空间。我们通常用字母 S 或希腊字母 Ω 来表示。对于一次公平的掷骰子，样本空间是 S = {1, 2, 3, 4, 5, 6}。了解样本空间是计算任何概率的第一步，因为它告诉我们等可能结果的总数。
4. Listing Outcomes | 列出结果
To ensure no outcome is missed, we can list outcomes systematically. For a single coin toss, the sample space is {Heads, Tails}. When two coins are flipped, we can list all pairs: HH, HT, TH, TT. Always use a fixed order to avoid duplicates, for example always write the first coin’s result then the second.
为了确保没有遗漏任何结果，我们可以系统地列出所有结果。对于一次抛硬币，样本空间是{正面，反面}。当抛两枚硬币时，我们可以列出所有组合：HH, HT, TH, TT。务必使用固定的顺序以避免重复，例如总是先写第一枚硬币的结果，再写第二枚。
5. Theoretical Probability Formula | 理论概率公式
When all outcomes in the sample space are equally likely, the probability of an event E is given by:
当样本空间中的所有结果等可能时，事件 E 的概率由下式给出：
P(E) = Number of favourable outcomes ÷ Total number of outcomes
For example, the probability of drawing a red card from a standard deck of 52 playing cards is 26/52, which simplifies to 1/2. This formula only works when every outcome has the same chance, such as with fair dice, spinners or coins.
例如，从一副标准的52张扑克牌中抽到一张红色牌的概率是26/52，化简为1/2。这个公式仅在每个结果机会均等时才有效，比如公平的骰子、转盘或硬币。
6. Probability Scale | 概率尺度
We can place the probability of any event on a number line from 0 to 1. An impossible event has probability 0, a certain event has probability 1, and an event with an even chance sits exactly in the middle at 0.5 or ½. Events that are unlikely have probabilities less than ½, while likely events have probabilities greater than ½.
我们可以把任何事件的概率放在从0到1的数轴上。不可能事件的概率为0，必然事件的概率为1，而机会均等的事件恰好位于中间，即0.5或½。不太可能的事件其概率小于½，而很可能的事件其概率大于½。
A probability close to 0, like 0.01, describes a very rare event; a probability close to 1, like 0.99, describes an almost certain event.
接近0的概率（如0.01）描述一个非常罕见的事件；接近1的概率（如0.99）描述一个几乎必然发生的事件。
7. Complementary Events | 互补事件
If E is an event, then the event that E does not happen is called the complement of E, often written as E’ or “not E”. Because either E or not E must happen, their probabilities add up to 1:
如果 E 是一个事件，那么 E 没有发生的事件称为 E 的补集，通常写作 E’ 或 “非 E”。由于 E 和非 E 两者必居其一，它们的概率之和为1：
P(not E) = 1 − P(E)
For instance, if the probability of rain tomorrow is 0.3, the probability it does not rain is 1 − 0.3 = 0.7. This relationship often saves time when it is easier to calculate the complement.
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。当计算补集的概率更容易时，这种关系常常能节省时间。
8. Experimental Probability vs Theoretical | 实验概率与理论概率
Theoretical probability tells us what we expect to happen in the long run based on equally likely outcomes. Experimental probability comes from actually carrying out an experiment and recording what happens:
理论概率基于等可能结果告诉我们在长远中预期会发生什么。实验概率则来自实际进行实验并记录发生的情况：
Experimental probability = Number of times the event occurs ÷ Total number of trials
If you toss a coin 100 times and get 47 heads, the experimental probability of heads is 47/100 = 0.47, whereas the theoretical probability is 0.5. The more trials you do, the closer the experimental probability usually gets to the theoretical probability.
如果你抛硬币100次得到47次正面，正面的实验概率是47/100 = 0.47，而理论概率是0.5。你进行的试验次数越多，实验概率通常越接近理论概率。
9. Using Sample Space Diagrams | 使用样本空间图
A sample space diagram shows every possible outcome of a two‑step experiment. For tossing two coins, the sample space can be shown as:
样本空间图展示了一个两步实验的每个可能结果。对于抛两枚硬币，样本空间可以表示为：
| HH | HT |
| TH | TT |
From this diagram you can see there are four equally likely outcomes. The probability of getting exactly one head is 2/4 = 1/2.
从这个图中你可以看到有四个等可能的结果。恰好得到一个正面的概率是2/4 = 1/2。
10. Two-Step Experiments and Tables | 两步实验与表格
When two separate events happen, for example rolling two dice, a table is a very efficient way to list the sample space. Below is the sample space for the sum of two dice, but you would list all 36 ordered pairs like (1,1), (1,2), … up to (6,6). A table helps answer questions such as “What is the probability that the sum is 7?”
当两个独立事件发生时，例如掷两个骰子，使用表格是列出样本空间的一种非常高效的方式。以下是两个骰子点数和的部分样本空间，但你应该列出所有36个有序对，如 (1,1), (1,2), … 直到 (6,6)。表格有助于回答诸如“和为7的概率是多少？”这样的问题。
| + | 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 |
The sum appears 6 times out of 36 possible outcomes, so P(sum = 7) = 6/36 = 1/6. Tables are especially helpful for experiments involving two independent dice, spinners or cards.
和为7在36个可能结果中出现6次，因此 P(和为7) = 6/36 = 1/6。对于涉及两个独立骰子、转盘或纸牌的实验，表格格外有用。
11. Probability of Combined Events (Simple) | 复合事件的概率（简单）
A combined event involves more than one single outcome. For example, when rolling a fair die, the event “rolling an odd number or a number greater than 4” includes the outcomes 1, 3, 5 and 6. Using the sample space, there are 4 favourable outcomes out of 6, so the probability is 4/6 = 2/3. As long as outcomes are equally likely, you can always count favourable cases and divide by the total.
复合事件包含不止一个单一结果。例如，当掷一个公平的骰子时，事件“掷出奇数或大于4的数”包含了结果1、3、5和6。利用样本空间，6个结果中有4个有利结果，因此概率是4/6 = 2/3。只要结果等可能，你总是可以数出有利情况的个数并除以总数。
When events are mutually exclusive (they cannot happen at the same time), you can add their individual probabilities. But be careful not to double‑count outcomes that belong to both events.
当事件互斥（它们不能同时发生）时，你可以把它们的单独概率相加。但要小心不要重复计算属于两个事件的结果。
12. Key Points Summary | 要点总结
Probability measures the chance of an event and is always a number between 0 and 1. The sample space lists all possible equally likely outcomes. The probability of an event is found by dividing the number of favourable outcomes by the total number of outcomes. Complementary events satisfy P(not E) = 1 − P(E). Using systematic lists, sample space diagrams and tables helps you see every outcome clearly, making probability calculations straightforward and reliable.
概率度量事件发生的机会，并且始终是0到1之间的一个数。样本空间列出所有等可能的可能结果。事件的概率通过将有利结果的个数除以结果总数来求得。互补事件满足 P(非 E) = 1 − P(E)。使用系统列表、样本空间图和表格可以帮助你清晰地看到每一个结果，从而使概率计算直接而可靠。
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