📚 Probability | 概率
Probability is the branch of mathematics that deals with measuring how likely events are to happen. In KS3 Cambridge Mathematics, you learn to use a scale from 0 to 1, calculate basic probabilities, work with sample spaces, and explore experimental data. This article covers the key ideas you will meet when studying probability, with clear examples and explanations.
概率是数学中衡量事件发生可能性的分支。在 KS3 剑桥数学中，你将学习使用从 0 到 1 的尺度、计算基本概率、处理样本空间以及探索实验数据。本文涵盖了学习概率时会遇到的关键概念，并配有清晰的示例和解释。
1. The Probability Scale | 概率尺度
Probability is always a number between 0 and 1 inclusive. A probability of 0 means an event is impossible, while a probability of 1 means it is certain. Values in between indicate increasing likelihood. We can also express probabilities as fractions, decimals, or percentages.
概率始终是 0 到 1（含）之间的一个数。概率为 0 表示事件不可能发生，概率为 1 表示事件必然发生。中间的值表示可能性递增。我们还可以用分数、小数或百分数表示概率。
For example, tossing a fair coin and getting heads has a probability of ½ (0.5 or 50%). Rolling a 7 on a standard six‑sided die is impossible, so the probability is 0. Rolling a number less than 7 is certain, giving a probability of 1.
例如，抛一枚均匀硬币得到正面的概率是 ½（0.5 或 50%）。在一个标准六面骰子上掷出 7 是不可能的，因此概率为 0。掷出一个小于 7 的数是必然的，概率为 1。
| 0 | ¼ | ½ | ¾ | 1 |
| Impossible | Unlikely | As likely as not | Likely | Certain |
| 不可能 | 不太可能 | 可能性各半 | 很可能 | 一定 |
2. Sample Space | 样本空间
The sample space is the set of all possible outcomes of an experiment. Listing outcomes systematically helps us count them correctly. For a single die roll, the sample space is {1,2,3,4,5,6}. For two coins tossed together, the sample space can be shown as {HH, HT, TH, TT}.
样本空间是某个试验所有可能结果的集合。系统地列出结果有助于我们正确计数。对于掷一个骰子，样本空间是 {1,2,3,4,5,6}。对于同时抛两枚硬币，样本空间可以表示为 {HH, HT, TH, TT}。
When combining events, a sample space diagram (such as a two-way table) is very useful. For example, when rolling two dice, a 6x6 table shows all 36 equally likely outcomes. Being able to construct and read such diagrams is an essential KS3 skill.
当组合多个事件时，样本空间图（如双向表）非常有用。例如，掷两个骰子时，一个 6×6 的表格能表示所有 36 个等可能结果。能够构建并阅读这类图表是 KS3 阶段的重要技能。
3. Calculating Basic Probability | 计算基本概率
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 的概率由下式给出：
P(A) = number of favourable outcomes / total number of outcomes
d and 2 blue counters, picking one counter at random gives P(red) = 3/5 and P(blue) = 2/5.
这个公式很直接，但你必须始终检查结果是否真的等可能。在一个装有 3 个红色和 2 个蓝色计数片的袋子里，随机摸出一个计数片，P(红色) = 3/5，P(蓝色) = 2/5。
Probabilities can be simplified, but it is often helpful to keep the denominator as the total number of outcomes to compare fractions. Remember: the sum of probabilities of all mutually exclusive outcomes in a sample space is 1.
概率可以化简，但保留分母为总结果数通常有助于比较分数。请记住：样本空间中所有互斥结果的概率之和为 1。
4. Probability of an Event NOT Occurring | 事件不发生的概率
The probability that event A does not happen is 1 minus the probability that it does happen. This is sometimes called the complement of A, written as A'.
事件 A 不发生的概率等于 1 减去它发生的概率。它有时被称为 A 的补集，记作 A'。
P(not A) = 1 - P(A)
.7. This rule is particularly useful when it is easier to calculate the probability of the opposite event.
如果明天下雨的概率是 0.3，那么不下雨的概率是 1 - 0.3 = 0.7。当计算对立事件的概率更容易时，这条规则尤其有用。
For example, to find the probability of rolling at least one six in two dice throws, it is simpler to find the probability of no sixes and subtract from 1.
例如，要求掷两次骰子至少出现一个 6 的概率，更简单的方法是先求出没有 6 的概率，再从 1 中减去。
5. Experimental vs Theoretical Probability | 实验概率与理论概率
Theoretical probability is what we expect to happen mathematically, based on equally likely outcomes. Experimental probability (or relative frequency) comes from actually performing an experiment or collecting data.
理论概率是我们基于等可能结果在数学上预期发生的概率。实验概率（或相对频率）来自实际进行试验或收集数据。
Experimental probability = number of times event occurs / total number of trials
retical probability remains 0.5. As the number of trials increases, experimental probability tends to get closer to theoretical probability.
如果你抛硬币 50 次，得到 28 次正面，正面的实验概率为 28/50 = 0.56。理论概率仍然是 0.5。随着试验次数增加，实验概率往往更接近理论概率。
This relationship is called the Law of Large Numbers. In KS3, you will often compare results from an experiment with theoretical expectations and discuss possible reasons for differences, such as bias or insufficient trials.
这种关系被称为大数定律。在 KS3，你将经常比较实验结果与理论预期，并讨论可能产生差异的原因，例如偏差或试验次数不足。
6. 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 die once, the events 'rolling a 3' and 'rolling a 5' are mutually exclusive. The addition rule for mutually exclusive events states:
如果两个事件不可能同时发生，则它们是互斥的。例如，掷一次骰子时，“掷出 3”和“掷出 5”是互斥的。关于互斥事件的加法规则如下：
P(A or B) = P(A) + P(B)
If the probability of drawing a red card from a well‑ lack card is ½ + ½ = 1 (certain). This rule works only when events are mutually exclusive.
如果从一副洗好的牌中抽到红牌的概率是 ½，抽到黑牌的概率也是 ½，那么抽到红牌或黑牌的概率是 ½ + ½ = 1（必然事件）。该规则仅在事件互斥时成立。
If events can overlap (like 'drawing a Queen' and 'drawing a red card'), they are not mutually exclusive and the addition rule requires an adjustment. At KS3 you will mostly work with mutually exclusive scenarios.
如果事件可能重叠（例如“抽到 Q”和“抽到红牌”），它们就不是互斥的，加法规则需要调整。在 KS3 阶段，你主要处理互斥的情况。
7. Independent Events | 独立事件
Independent events are those where the outcome of one does not affect the outcome of the other. For example, tossing a coin and rolling a die are independent. The multiplication rule for independent events A and B is:
独立事件是指一个事件的结果不影响另一个事件的结果。例如，抛硬币和掷骰子是独立的。关于独立事件 A 和 B 的乘法规则是：
P(A and B) = P(A) × P(B)
the probability of rolling a 6 on a fair die is 1/6, then P(head and 6) = ½ × 1/6 = 1/12.
如果得到正面的概率是 ½，在一个均匀骰子上掷出 6 的概率是 1/6，那么 P(正面且 6) = ½ × 1/6 = 1/12。
Be careful: this rule only works for independent events. If events are dependent (like picking two counters from a bag without replacement), the probabilities change after the first pick. At KS3, you will meet both independent and dependent contexts through tree diagrams.
请小心：这条规则仅适用于独立事件。如果事件是相关的（比如从一个袋子中不替换地取出两个计数片），概率会在第一次取出后改变。在 KS3，你将通过树状图遇到独立和相关两种情形。
8. Tree Diagrams | 树状图
A tree diagram is a visual way of showing all possible outcomes of two or more events, especially when events are combined. Each branch is labelled with its probability. To find the probability of a specific combined outcome, you multiply along the branches.
树状图是展示两个或多个事件所有可能结果的可视化方法，尤其在组合事件时。每条分支标记其概率。要得出特定组合结果的概率，你只需沿分支相乘。
For example, a bag contains 3 green and 7 red sweets. If two sweets are taken at random, a tree diagram can show the outcomes with and without replacement. The probabilities on the second set of branches change in the without‑replacement case, because the totals decrease.
例如，一个袋子里有 3 颗绿色糖果和 7 颗红色糖果。如果随机取出两颗糖果，树状图可以表示有放回和无放回两种情况的结果。在无放回的情况下，第二组分枝上的概率会发生变化，因为总数减少。
KS3 questions often ask you to complete partly drawn tree diagrams or to use them to calculate probabilities such as P(at least one green). Always check that the sum of probabilities on branches from any point equals 1.
KS3 的题目经常会要求你补充残缺的树状图，或利用它们计算概率，如 P(至少一颗绿色)。始终要检查从任意一点出发的分支概率之和等于 1。
9. Relative Frequency and Expectation | 相对频率与期望
Relative frequency is another name for experimental probability. It is calculated after an experiment and can be used to estimate probabilities when theoretical calculation is not possible.
相对频率是实验概率的另一个名称。它在试验后计算得出，并可在无法进行理论计算时用于估计概率。
Relative frequency = frequency of event / total frequency
Expected frequency, on the other hand, uses the theoretical probability to predict how many times an event should occur in a number of trials:
另一方面，期望频数利用理论概率来预测某个事件在多次试验中应该发生的次数：
Expected frequency = probability × number of trials
0.4 × 200 = 80 times. This does not mean it will land exactly 80 times, but it is the long‑term average.
如果一个偏心转盘停在蓝色区域的概率是 0.4，你转动它 200 次，你期望它停在蓝色区域 0.4 × 200 = 80 次。这并不意味着恰好是 80 次，而是长期的平均值。
10. Systematic Listing and Fairness | 系统列举与公平性
To find all possible outcomes without missing or repeating any, systematic listing is crucial. Use ordered lists, tables, or Venn diagrams. For instance, listing all outcomes when two coins are tossed can be done by fixing the first outcome: H then H; H then T; T then H; T then T.
要找出所有可能结果且不遗漏不重复，系统列举至关重要。可使用有序列表、表格或韦恩图。例如，列出两枚硬币抛出的所有结果时，可以先固定第一个结果：H后H；H后T；T后H；T后T。
A game is fair if every player has the same probability of winning. In KS3 you might be asked to decide if a game using a spinner or dice is fair, and to adjust rules so that it becomes fair. This links probability to real‑life decision making.
如果每个玩家获胜的概率相同，则游戏是公平的。在 KS3 中，你可能会被要求判断一个使用转盘或骰子的游戏是否公平，并调整规则使其变得公平。这联系了概率与现实决策。
11. Probability in Everyday Contexts | 日常语境中的概率
Probability is used in weather forecasting, insurance, sports, and medicine. When a weather report says '70% chance of rain', it means that under similar conditions, rain has occurred 70 times out of 100. Understanding probability helps you interpret risks and make informed decisions.
概率被用于天气预报、保险、体育和医学等领域。当天气报告说“70% 的降雨概率”，是指在类似条件下，每 100 次中有 70 次发生了降雨。理解概率有助于你解读风险并做出明智决定。
In KS3, you may explore how probability affects everyday situations, such as the fairness of a raffle or the chance of winning a simple prize draw. These explorations build your ability to reason logically about uncertainty.
在 KS3 中，你可能会探索概率如何影响日常情境，例如抽奖活动的公平性或赢得简单抽奖的机会。这些探索能培养你关于不确定性的逻辑推理能力。
12. Summary and Key Tips | 总结与关键提示
Always check that probabilities lie between 0 and 1. Be certain whether events are mutually exclusive or independent before applying addition or multiplication rules. Use a sample space diagram when outcomes are equally likely, and a tree diagram for multi‑stage events. Remember the complement rule can save time. With experimental probability, more trials give more reliable estimates.
始终检查概率是否落在 0 和 1 之间。在应用加法或乘法规则之前，先确认事件是互斥的还是独立的。当结果等可能时使用样本空间图，对多阶段事件使用树状图。记住补集规则可以节省时间。对于实验概率，更多的试验次数会给出更可靠的估计。
Mastering these concepts will give you a strong foundation for IGCSE and beyond. Probability is a powerful tool for describing the world around us.
掌握这些概念将为你 IGCSE 及以后的学习奠定坚实基础。概率是描述我们周围世界的有力工具。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图