📚 Probability Basics | 概率基础
Probability is the branch of mathematics that helps us understand and measure uncertainty. Whether you are rolling a die, flipping a coin, or predicting the weather, probability gives us a way to describe how likely events are. In KS3 Cambridge Mathematics, you will learn to work with probabilities as fractions, decimals, and percentages, and to explore single events, combined events, and the difference between theoretical and experimental results.
概率是数学中帮助我们理解和度量不确定性的分支。无论你是在掷骰子、抛硬币还是预测天气，概率都为我们提供了一种描述事件发生可能性的方法。在 KS3 剑桥数学中，你将学习用分数、小数和百分比处理概率，并探究单一事件、组合事件以及理论结果与实验结果之间的差异。
1. Introduction: What is Probability? | 概率是什么？
Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means an event is impossible and 1 means it is certain. In everyday language we use terms like 'likely', 'unlikely', or 'even chance', but mathematics assigns precise numerical values to these ideas.
概率是衡量事件发生可能性的度量。它用 0 到 1 之间的数字表示，0 表示事件不可能发生，1 表示事件一定发生。在日常用语中，我们使用“很可能”“不太可能”或“均等机会”等词语，但数学为这些概念赋予了精确的数值。
2. Probability Scale | 概率标尺
The probability scale is a continuous line from 0 to 1. An event with probability 0 is impossible, such as rolling a 7 on a standard six-sided die. An event with probability 1 is certain, for example, the sun rising tomorrow morning. A probability of 0.5 represents an even chance, like getting heads when you toss a fair coin. Values close to 0 describe very unlikely events, while values near 1 indicate very likely ones.
概率标尺是从 0 到 1 的一条连续线段。概率为 0 的事件是不可能的，例如在一个标准六面骰子上掷出 7 点。概率为 1 的事件是确定的，例如太阳明天早上会升起。概率为 0.5 表示均等机会，如抛一枚公平硬币得到正面。接近 0 的值描述非常不可能的事件，而接近 1 的值表示非常可能发生的事件。
3. Basic Probability Formula | 基本概率公式
When all outcomes are equally likely, the probability of an event A happening is given by the formula:
当所有结果出现的可能性相同时，事件 A 发生的概率由以下公式给出：
P(A) = Number of favourable outcomes ÷ Total number of possible outcomes
For example, if you pick a card at random from a standard deck of 52 playing cards, the probability of drawing an ace is 4/52, which simplifies to 1/13. The formula works because there are 4 aces among 52 equally likely cards.
例如，如果你从一副标准的 52 张扑克牌中随机抽取一张，抽到 A 的概率是 4/52，化简后为 1/13。这个公式成立的原因是在 52 张等可能的牌中有 4 张 A。
4. Single Events: Coin Toss | 单一事件：抛硬币
Tossing a fair coin produces two equally likely outcomes: heads (H) and tails (T). The probability of getting heads is 1/2, and the probability of tails is also 1/2. Since each toss is independent, the result of one flip does not influence the next. This means that even after getting five heads in a row, the chance of heads on the next toss remains exactly 1/2.
抛一枚公平硬币会产生两种等可能的结果：正面 (H) 和反面 (T)。得到正面的概率是 1/2，得到反面的概率也是 1/2。由于每次抛掷是独立的，一次的结果不会影响下一次。这意味着即使连续得到五次正面，下一次抛掷得到正面的概率仍然是 1/2。
5. Single Events: Rolling a Die | 单一事件：掷骰子
3. Listing the outcomes that make up an event helps you count favourable cases accurately.
一个公平的六面骰子的可能结果为 {1, 2, 3, 4, 5, 6}。掷出偶数（即集合 {2, 4, 6}）的概率是 3/6 = 1/2。掷出大于 4 的数（即 {5, 6}）的概率是 2/6 = 1/3。列出构成事件的结果有助于你准确计数有利情形。
6. Probability as Fractions, Decimals and Percentages | 分数、小数和百分比的概率表示
Probability can be expressed in three equivalent forms: as a fraction, a decimal, or a percentage. For instance, a probability of 1/4 is the same as 0.25 and 25%. To change a fraction to a decimal, divide the numerator by the denominator. Multiply the decimal by 100 to turn it into a percentage. Being comfortable converting between these forms is essential in KS3 work and real-life contexts.
概率可以用三种等价形式表达：分数、小数或百分比。例如，1/4 的概率等同于 0.25 和 25%。要将分数转换为小数，用分子除以分母。再将小数乘以 100 就变成百分比。熟练地在这些形式之间转换对 KS3 的学习和实际生活都很重要。
7. Complementary Events | 互补事件
The complement of an event A, written as A' or 'not A', consists of all outcomes where A does not happen. Because it is certain that either A happens or it does not, their probabilities add up to 1. This gives the very useful rule:
事件 A 的互补事件，记作 A' 或“非 A”，包含 A 不发生的所有结果。因为要么 A 发生，要么 A 不发生，这是确定的，所以它们的概率之和为 1。由此得出一个非常有用的规则：
P(not A) = 1 – P(A)
le, if the probability that a bus is on time is 0.85, then the probability it is late is 1 – 0.85 = 0.15. This concept helps you quickly find the chance of an event failing to occur.
例如，如果公共汽车准时的概率是 0.85，那么它晚点的概率就是 1 – 0.85 = 0.15。这个概念可以帮助你快速求出某事件不发生的概率。
8. Mutually Exclusive Events | 互斥事件
Two events are called mutually exclusive if they cannot occur at the same time. For a single roll of a die, 'rolling a 2' and 'rolling an odd number' are mutually exclusive because you cannot roll both. For mutually exclusive events A and B, the probability that either A or B happens is simply the sum of their individual probabilities:
如果两个事件不能同时发生，则称它们为互斥事件。对于掷一次骰子，“掷出 2”和“掷出奇数”就是互斥事件，因为你不可能同时掷出两者。对于互斥事件 A 和 B，A 或 B 发生的概率就是它们各自概率之和：
P(A or B) = P(A) + P(B)
If the events are not mutually exclusive, you must subtract the overlap to avoid counting it twice, but at KS3 you will mostly work with mutually exclusive cases.
如果事件不是互斥的，你必须减去重叠部分以避免重复计算，不过在 KS3 阶段你大多处理的是互斥事件的情形。
9. Experimental vs Theoretical Probability | 实验概率与理论概率
al probability of heads is 47/100 = 0.47, whereas the theoretical probability is 0.5. The Law of Large Numbers tells us that as you increase the number of trials, the experimental probability will tend to move closer to the theoretical value.
理论概率是我们基于等可能结果所预期的情况。实验概率（也称相对频率）则来自于实际进行实验或观察试验。例如，如果你抛硬币 100 次得到 47 次正面，正面的实验概率就是 47/100 = 0.47，而理论概率是 0.5。大数定律告诉我们，当试验次数增加时，实验概率会趋近于理论值。
10. Sample Space Diagrams | 样本空间图示
. For example, the sum 7 appears in six cells, so P(sum = 7) = 6/36 = 1/6. Using a table helps you visualise all possibilities and count favourable outcomes accurately.
样本空间是实验所有可能结果的集合。抛两枚硬币时，样本空间可以系统性地列出：{HH, HT, TH, TT}。当组合两个独立事件时，二维表格是一个强有力的工具。对于掷两个骰子，一个 6×6 的表格展示了 36 种等可能的结果。例如，和为 7 出现在六个单元格中，因此 P(和为 7) = 6/36 = 1/6。使用表格有助于你将所有可能性可视化并准确计数有利结果。
11. Probability Trees for Independent Events | 独立事件的概率树
ts into H (½) and T (½), and from each of those the second toss again branches H (½) and T (½). The probability of getting two heads is ½ × ½ = ¼. You can then add probabilities of different paths when they lead to the same final outcome.
树形图非常适合展示一系列独立事件，每条分支都标有结果及其概率。要计算一条完整路径的概率，你需要将沿途分支上的概率相乘。例如，抛一枚公平硬币两次：第一条分支分为 H (½) 和 T (½)，然后从这两条分支出发，第二次抛掷又各自分出 H (½) 和 T (½)。得到两个正面的概率是 ½ × ½ = ¼。当不同路径通往相同最终结果时，你可以将它们的概率相加。
12. Real-life Applications | 实际应用
Probability is used extensively in many fields: weather forecasters use it to predict rain, insurance companies calculate premiums based on risk, and game designers balance winning odds. In KS3, you will build a strong foundation by learning to calculate probabilities for single events and simple combined events, using fractions, decimals, percentages, sample spaces and tree diagrams. Mastering these skills will help you make better decisions based on data and likelihoods in everyday life.
概率在许多领域有着广泛的应用：天气预报员用它预测降雨，保险公司基于风险计算保费，游戏设计者用它平衡获胜几率。在 KS3 阶段，你将通过计算单一事件和简单组合事件的概率，使用分数、小数、百分比、样本空间和树形图，打下坚实的基础。掌握这些技能将帮助你在日常生活中依据数据和可能性做出更好的决策。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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