📚 Basic Probability | 基础概率
Probability is the branch of mathematics that deals with chance and uncertainty. When you flip a coin, roll a die, or pick a card from a deck, you are performing an experiment with uncertain outcomes. Understanding probability helps you predict how likely events are to happen, from everyday situations to scientific experiments.
概率是数学中研究机会和不确定性的分支。当你抛硬币、掷骰子或从一副牌中抽牌时，你正在执行一个结果不确定的实验。理解概率可以帮助你预测事件发生的可能性，无论是日常生活还是科学实验。
1. What is Probability? | 什么是概率？
Probability describes the likelihood that a particular event will occur. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. An outcome is a possible result of an experiment, and an event is a set of one or more outcomes.
概率描述了某个特定事件发生的可能性。它用一个介于 0 和 1 之间的数字表示，0 表示事件不可能发生，1 表示事件必然发生。结果是实验的可能结果，事件是一个或多个结果的集合。
For example, when you roll a fair six-sided die, the outcomes are 1, 2, 3, 4, 5, 6. The event “rolling an even number” includes the outcomes 2, 4, 6.
例如，掷一个公平的六面骰子时，结果是 1、2、3、4、5、6。事件“掷出偶数”包括结果 2、4、6。
2. The Probability Scale | 概率尺度
The probability scale is a visual way to show how likely an event is. It runs from 0 (impossible) to 1 (certain). Words like ‘unlikely’, ‘even chance’, ‘likely’ can be placed on the scale. An even chance is exactly ½ or 0.5.
概率尺度是一种直观表示事件可能性的方法。范围从 0（不可能）到 1（必然）。像“不太可能”、“机会均等”、“很可能”等词语可以放在尺度上。机会均等恰好是 ½ 或 0.5。
We can represent the probability scale as a line with markings:
我们可以用带有标记的线段表示概率尺度：
0 ——— ½ ——— 1
3. Outcomes and Sample Space | 结果与样本空间
The sample space is the set of all possible outcomes of an experiment. For tossing a coin, the sample space is {Heads, Tails}. For rolling a die, it is {1, 2, 3, 4, 5, 6}. Listing outcomes systematically helps ensure none are missed.
样本空间是实验中所有可能结果的集合。抛一枚硬币时，样本空间是 {正面, 反面}。掷一个骰子时，样本空间是 {1, 2, 3, 4, 5, 6}。系统地列出结果有助于确保没有遗漏。
When rolling two dice, you can use a table to show all 36 outcomes. The sample space size is the total number of outcomes.
掷两个骰子时，你可以使用表格显示所有 36 种结果。样本空间的大小就是结果的总数。
4. Calculating Probability | 计算概率
If all outcomes are equally likely, the probability of an event A is given by:
如果所有结果等可能发生，事件 A 的概率由下式计算：
P(A) = Number of favourable outcomes / Total number of outcomes
For a fair die, P(rolling a 3) = ⅙. P(rolling an even number) = 3/6 = ½. Probability can be written as a fraction, decimal, or percentage.
对于一个公平的骰子，P(掷出 3) = ⅙。P(掷出偶数) = 3/6 = ½。概率可以写成分数、小数或百分比。
5. Equally Likely Outcomes | 等可能结果
Equally likely outcomes mean that every outcome has the same chance of occurring. A fair coin, a fair die, and a well-shuffled deck of cards all produce equally likely outcomes. When outcomes are not equally likely (e.g., a biased die), the simple formula does not apply directly; instead, we use relative frequency.
等可能结果意味着每个结果发生的可能性相同。公平的硬币、公平的骰子和洗匀的扑克牌都产生等可能的结果。当结果不是等可能时（例如，有偏骰子），不能直接使用简单公式；此时应使用相对频率。
It is important to check for fairness before assuming equally likely outcomes.
在假设结果等可能之前，检查公平性很重要。
6. Probability of an Event Not Happening | 事件不发生的概率
The probability that an event A does not happen is called the complement of A, denoted A’. Since the total probability sums to 1, we have:
事件 A 不发生的概率称为 A 的补集，记作 A’。由于总概率之和为 1，我们有：
P(A’) = 1 − P(A)
If the probability of rain tomorrow is ¼, then the probability it does not rain is 1 − ¼ = ¾.
如果明天下雨的概率是 ¼，那么不下雨的概率是 1 − ¼ = ¾。
7. Experimental Probability | 实验概率
Experimental probability (also called relative frequency) is based on actual experiments or historical data. It is calculated as:
实验概率（也叫相对频率）基于实际实验或历史数据。计算公式为：
Experimental probability = Number of times event occurs / Total number of trials
For example, if you flip a coin 100 times and get 47 heads, the experimental probability of heads is 47/100 = 0.47.
例如，如果你抛硬币 100 次得到 47 次正面，则正面的实验概率是 47/100 = 0.47。
8. Comparing Theoretical and Experimental Probability | 比较理论概率与实验概率
Theoretical probability is what we expect to happen, while experimental probability is what actually happens. As the number of trials increases, the experimental probability usually gets closer to the theoretical probability. This is known as the law of large numbers.
理论概率是我们期望发生的情况，而实验概率是实际发生的情况。随着试验次数的增加，实验概率通常会趋近理论概率。这被称为大数定律。
If you flip a fair coin many times, the experimental probability of heads should approach ½.
如果你多次抛掷一枚公平的硬币，正面的实验概率应趋近于 ½。
9. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot both happen at the same time. For example, when rolling a die, getting a 2 and getting a 5 are mutually exclusive because a single roll cannot show both 2 and 5. The probability of either one or the other occurring is the sum of their individual probabilities:
如果两个事件不能同时发生，则它们是互斥的。例如，掷骰子时，掷出 2 和掷出 5 是互斥的，因为一次掷骰不能同时显示 2 和 5。其中一个或另一个发生的概率等于它们各自概率之和：
P(A or B) = P(A) + P(B)
Thus, P(2 or 5) = ⅙ + ⅙ = ⅓.
因此，P(2 或 5) = ⅙ + ⅙ = ⅓。
10. Exhaustive Events | 穷举事件
Exhaustive events are a set of events that cover all possible outcomes. Together, their probabilities sum to 1. For a die, the events “rolling an even number” and “rolling an odd number” are mutually exclusive and exhaustive.
穷举事件是涵盖所有可能结果的一组事件。它们的概率之和为 1。对于骰子，“掷出偶数”和“掷出奇数”这两个事件是互斥且穷举的。
When events are exhaustive and mutually exclusive, P(A) + P(B) = 1, and A and B are complementary.
当事件穷举且互斥时，P(A) + P(B) = 1，且 A 和 B 互补。
11. Two-Way Tables and Probability | 双向表与概率
Two-way tables help organise outcomes for two combined events. For instance, a table can show the sum of two dice. Each cell represents a combined outcome. We can use the table to calculate probabilities like P(sum = 7).
双向表有助于整理两个组合事件的结果。例如，一个表格可以显示两个骰子的点数之和。每个单元格代表一个组合结果。我们可以使用表格计算概率，如 P(和为 7)。
In a 6×6 table, sums range from 2 to 12. The number of ways to get a sum of 7 is 6, out of 36. So P(sum = 7) = ⅙.
在一个 6×6 表格中，和的范围从 2 到 12。得到和为 7 的方式有 6 种，在 36 种中。所以 P(和为 7) = ⅙。
Let’s construct a small table as an example:
让我们构建一个小表格作为示例：
| + | 1 | 2 | 3 |
| 1 | 2 | 3 | 4 |
| 2 | 3 | 4 | 5 |
| 3 | 4 | 5 | 6 |
12. Tree Diagrams (Introduction) | 树状图（简介）
A tree diagram is a way to show all possible outcomes of two or more experiments. Each branch represents a possible outcome and its probability. For two coin flips, you can draw a tree to see {HH, HT, TH, TT}. The probability of each final outcome is found by multiplying probabilities along the branches.
树状图是一种显示两个或多个实验所有可能结果的方法。每个分支代表一个可能的结果及其概率。对于两次抛硬币，你可以画一棵树来显示 {正正, 正反, 反正, 反反}。每个最终结果的概率通过沿分支上的概率相乘得到。
If the coin is fair, each branch has probability ½, so P(HH) = ½ × ½ = ¼. Tree diagrams are useful when events are independent.
如果硬币公平，每个分支的概率为 ½，因此 P(正正) = ½ × ½ = ¼。当事件独立时，树状图非常有用。
Published by TutorHao | Mathematics Revision Series | aleveler.com
Find Maths Textbooks on eBay UK
New, used and second-hand copies of textbooks and revision guides are often much cheaper than retail — check current listings and prices before you buy.
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导