📚 Probability: From Impossible to Certain | 概率：从不可能到必然
Probability is the branch of mathematics that deals with chance and uncertainty. It allows us to measure how likely an event is to happen, ranging from impossible events to certain events. In KS3 Cambridge Mathematics, probability is introduced through probability language, the probability scale, fractions, decimals, percentages, sample space diagrams and experimental data.
概率是数学中处理偶然性和不确定性的分支。它使我们能够衡量一个事件发生的可能性，范围从不可能事件到必然事件。在 KS3 剑桥数学中，概率通过概率语言、概率尺度、分数、小数、百分数、样本空间图和实验数据等内容引入。
1. What is Probability? | 什么是概率？
Probability gives a numerical measure of how likely an event is to occur. It can be written as a fraction, a decimal or a percentage. For example, the probability of a fair coin landing heads up is 1/2, 0.5 or 50%.
概率给出事件发生可能性的数值度量。它可以用分数、小数或百分数表示。例如，一枚公平硬币正面朝上的概率是 1/2、0.5 或 50%。
We use probability in everyday life, from weather forecasts to games and risk assessments. Understanding how to calculate probability helps us make informed decisions when outcomes are uncertain.
我们在日常生活中使用概率，从天气预报到游戏和风险评估。理解如何计算概率有助于我们在结果不确定时做出明智的决定。
2. The Probability Scale | 概率尺度
The probability scale goes from 0 to 1. A probability of 0 means an event is impossible, and a probability of 1 means an event is certain.
概率尺度从 0 到 1。概率为 0 表示事件不可能，概率为 1 表示事件必然发生。
A probability close to 0 means very unlikely, while a probability close to 1 means very likely. The midpoint 1/2 means an even chance, where an event is just as likely to happen as not to happen.
接近 0 的概率表示非常不可能，接近 1 的概率表示非常可能。中点 1/2 表示均等机会，即事件发生和不发生的可能性相同。
- 0 – impossible (不可能)
- 1/4 – unlikely (不太可能)
- 1/2 – even chance (均等机会)
- 3/4 – likely (很可能)
- 1 – certain (必然发生)
3. Basic Probability Formula | 基本概率公式
To find the probability of an event A, use the formula below.
要计算事件 A 的概率，使用以下公式。
P(A) = number of favourable outcomes ÷ total number of outcomes
e, if a bag contains 4 red and 6 blue balls, the probability of picking red is 4/10 = 2/5.
例如，如果一个袋子里有 4 个红球和 6 个蓝球，取出红球的概率是 4/10 = 2/5。
The outcome must be equally likely for this formula to be valid. If some outcomes are more likely than others, the simple ratio no longer applies directly.
要使这个公式有效，所有结果必须是等可能的。如果某些结果比其他结果更可能发生，简单比值就不再直接适用。
4. Probability of an Event Not Happening | 事件不发生的概率
If P(A) is the probability that event A happens, then the probability that A does not happen is given by the complement rule.
如果 P(A) 是事件 A 发生的概率，那么事件 A 不发生的概率由补集规则给出。
P(not A) = 1 − P(A)
, if the probability of rain tomorrow is 0.3, the probability of no rain is 1 − 0.3 = 0.7.
例如，如果明天下雨的概率是 0.3，那么不下雨的概率是 1 − 0.3 = 0.7。
This rule is very useful because sometimes it is easier to find the probability of the opposite event and subtract from 1.
这条规则非常有用，因为有时先求相反事件的概率再用 1 减去会更简单。
5. Sample Space Diagrams | 样本空间图
A sample space is the set of all possible outcomes of an experiment. A sample space diagram helps to organise outcomes for two events, such as rolling two dice.
样本空间是一个实验所有可能结果的集合。样本空间图有助于整理两个事件的结果，例如掷两个骰子。
6 = 36 equally likely outcomes. We can list these in a table to see combinations clearly.
对于两个公平骰子，样本空间有 6 × 6 = 36 个等可能结果。我们可以用表格列出这些结果，以清楚地看到组合。
| + | 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 |
ossible sums when two dice are rolled. This makes it easy to find P(sum = 7) = 6/36 = 1/6.
该表显示了掷两个骰子时所有可能的和。这样可以轻松求出 P(和为 7) = 6/36 = 1/6。
6. Mutually Exclusive Events | 互斥事件
Mutually exclusive events cannot happen at the same time. For example, when rolling a die, the events 'rolling an even number' and 'rolling a 5' are mutually exclusive.
互斥事件不能同时发生。例如，掷骰子时，“掷出偶数”和“掷出 5”这两个事件是互斥的。
If two events A and B are mutually exclusive, the probability of A or B happening is the sum of their individual probabilities.
如果两个事件 A 和 B 互斥，那么 A 或 B 发生的概率是它们各自概率的和。
P(A or B) = P(A) + P(B)
This rule is used when asking for a combined probability of separate outcomes that cannot overlap.
这个规则用于求不会重叠的独立结果的组合概率。
7. Experimental Probability vs Theoretical Probability | 实验概率与理论概率
Theoretical probability is calculated by assuming all outcomes are equally likely. Experimental probability is based on actual trials or experiments.
理论概率是假设所有结果等可能时计算出来的。实验概率则基于实际试验或实验。
s found using the formula P(event) = number of successful trials ÷ total number of trials.
实验概率使用公式 P(事件) = 成功试验次数 ÷ 总试验次数。
As the number of trials increases, experimental probability usually gets closer to theoretical probability. With a small number of trials, results can vary a lot.
随着试验次数增加，实验概率通常越来越接近理论概率。试验次数较少时，结果可能会有很大波动。
8. Relative Frequency | 相对频率
Relative frequency is another name for experimental probability. It is the ratio of the frequency of an event to the total number of trials.
相对频率是实验概率的另一个名称。它是事件发生的频数与总试验次数之比。 Published by TutorHao | KS3 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