Mastering Basic Probability | 掌握基础概率

📚 Mastering Basic Probability | 掌握基础概率

Probability is one of the most useful and practical topics in KS3 Cambridge Mathematics. It helps us to describe how likely something is to happen, from the roll of a dice to the chance of rain tomorrow. In this article, we will build a clear understanding of basic probability, learn how to calculate it, avoid common mistakes, and prepare confidently for Cambridge checkpoint questions.

概率是 KS3 剑桥数学中最实用、最贴近生活的主题之一。它帮助我们描述事件发生的可能性大小，从掷骰子到明天是否下雨。在这篇文章中，我们将系统地理解基础概率，学会计算概率，避开常见错误，并为剑桥 checkpoint 考试做好充分准备。

1. What Is Probability? | 什么是概率？

Probability is a measure of how likely an event is to happen. It is used in weather forecasts, games, insurance, and everyday risk assessment. An event can be something like ‘rolling a six’ or ‘picking a red ball from a bag’.

概率是衡量事件发生可能性大小的度量。它广泛用于天气预报、游戏、保险和日常风险评估中。事件可以是“掷出 6 点”或“从袋子中取出红球”等情况。

In mathematics, probability gives us a numerical value for this likelihood. The higher the probability, the more likely the event is to happen.

在数学中，概率为这种可能性赋予一个数值。概率越高，事件发生的可能性就越大。

2. The Probability Scale | 概率标尺

All probabilities lie between 0 and 1, inclusive. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain.

所有概率都在 0 到 1 之间（含 0 和 1）。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。

0 = impossible（不可能）
0.25 = unlikely（不太可能）
0.5 = even chance（机会均等）
0.75 = likely（很可能）
1 = certain（必然发生）

We can also write probabilities as fractions, decimals, or percentages. For example, an even chance can be written as 1/2, 0.5, or 50%.

概率还可以用分数、小数或百分数表示。例如，机会均等可以写成 1/2、0.5 或 50%。

3. Sample Space and Outcomes | 样本空间与结果

A sample space is the set of all possible outcomes of an experiment. For a fair coin, the sample space is {Heads, Tails}. For a fair six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.

样本空间是指一次试验中所有可能结果的集合。对于一枚均匀硬币，样本空间为 {正面, 反面}。对于一枚均匀六面骰子，样本空间为 {1, 2, 3, 4, 5, 6}。

Listing the sample space clearly is the first step in solving many probability problems. For example, when two coins are tossed, the sample space has four outcomes: HH, HT, TH, TT.

清晰列出样本空间是解决许多概率问题的第一步。例如，抛掷两枚硬币时，样本空间有四种结果：HH、HT、TH、TT。

4. Equally Likely Outcomes | 等可能结果

Equally likely outcomes are outcomes that have the same chance of occurring. When a fair die is rolled, each of the six numbers is equally likely, so each has a probability of 1/6.

等可能结果是指发生机会相同的结果。掷一枚均匀骰子时，六个数字中每个数出现的可能性相同，因此每个数字的概率都是 1/6。

When all outcomes are equally likely, we can calculate theoretical probability by counting. This is the most common type of probability question in KS3.

当所有结果等可能时，我们可以通过计数来计算理论概率。这是 KS3 阶段最常见的概率题型。

5. Theoretical Probability Formula | 理论概率公式

When all outcomes are equally likely, the probability of an event A is given by the following formula.

当所有结果等可能时，事件 A 的概率由以下公式给出。

P(A) = 有利结果数 ÷ 总结果数

For example, a fair die has six possible outcomes. The event ‘rolling an odd number’ has three favourable outcomes: 1, 3, and 5. Therefore, P(odd) = 3/6 = 1/2.

例如，一枚均匀骰子有六种可能结果。事件“掷出奇数”有三个有利结果：1、3 和 5。因此，P(奇数) = 3/6 = 1/2。

Always check whether the fraction can be simplified. A probability should usually be given in its simplest form unless the question states otherwise.

一定要检查分数是否可以约分。除非题目另有说明，概率通常应写成最简形式。

6. Complementary Events | 互补事件

The complement of an event A is the event that A does not happen. It is often written as A’ or not A. The probabilities of an event and its complement always add up to 1.

事件 A 的互补事件是指 A 不发生的事件，通常记作 A’ 或 not A。一个事件与其互补事件的概率之和总是等于 1。

P(not A) = 1 − P(A)

For example, if the probability of rolling a six is 1/6, then the probability of not rolling a six is 1 − 1/6 = 5/6. This rule is very useful for checking answers.

例如，如果掷出 6 点的概率是 1/6，那么没有掷出 6 点的概率就是 1 − 1/6 = 5/6。这个规则在检查答案时非常有用。

7. Mutually Exclusive Events and Addition | 互斥事件与加法法则

Mutually exclusive events are events that cannot happen at the same time. For example, when rolling a die, ‘rolling a 2’ and ‘rolling a 5’ are mutually exclusive because you cannot get both on one roll.

互斥事件是指不能同时发生的事件。例如，掷骰子时，’掷出 2 点’和’掷出 5 点’是互斥事件，因为一次掷骰不可能同时得到两个点数。

For mutually exclusive events A and B, the probability of A or B happening is the sum of their individual probabilities.

对于互斥事件 A 和 B，A 或 B 发生的概率等于它们各自概率之和。

P(A or B) = P(A) + P(B)

For example, a bag contains 3 red, 2 blue and 5 green balls. The probability of picking a red ball is 3/10, and the probability of picking a blue ball is 2/10. Therefore, P(red or blue) = 3/10 + 2/10 = 5/10 = 1/2.

例如，一个袋子里有 3 个红球、2 个蓝球和 5 个绿球。取出红球的概率是 3/10，取出蓝球的概率是 2/10。因此，P(红或蓝) = 3/10 + 2/10 = 5/10 = 1/2。

8. Experimental Probability and Relative Frequency | 实验概率与相对频率

Experimental probability is based on actual trials or data. It is often called relative frequency. The relative frequency of an event is calculated as the number of successful trials divided by the total number of trials.

实验概率基于实际试验或数据，通常称为相对频率。事件的相对频率等于成功试验的次数除以总试验次数。

相对频率 = 成功次数 ÷ 总试验次数

For example, a spinner is spun 50 times and lands on red 18 times. The experimental probability of landing on red is 18/50 = 9/25.

例如，一个转盘转动 50 次，其中 18 次停在红色区域。停在红色区域的实验概率为 18/50 = 9/25。

As the number of trials increases, the experimental probability usually gets closer to the theoretical probability. This is called the law of large numbers.

随着试验次数的增加，实验概率通常会越来越接近理论概率。这称为大数定律。

9. Expected Frequency | 期望频数

Expected frequency predicts how many times an event should occur in a given number of trials. It is calculated by multiplying the probability of the event by the number of trials.

期望频数用于预测事件在给定试验次数中应发生的次数。它等于事件发生的概率乘以试验次数。

期望频数 = P(A) × 试验次数

For example, a fair die is rolled 300 times. The probability of rolling a six is 1/6, so the expected frequency of rolling a six is 1/6 × 300 = 50 times.

例如，一枚均匀骰子被掷 300 次。掷出 6 点的概率是 1/6，所以掷出 6 点的期望频数为 1/6 × 300 = 50 次。

Remember that expected frequency is a prediction based on probability, not a guarantee of the actual outcome in an experiment.

请记住，期望频数是基于概率的预测，并不保证试验的实际结果恰好如此。

10. Listing Outcomes and Tree Diagrams | 列出结果与树状图

For combined events, such as tossing two coins or rolling two dice, we need a systematic way to list all possible outcomes. A tree diagram is a useful tool for showing all possible combinations.

对于组合事件，例如抛两枚硬币或掷两个骰子，我们需要一种系统的方法列出所有可能结果。树状图是展示所有可能组合的有用工具。

For example, when tossing two coins, the tree diagram shows two branches for the first coin (H or T) and then two branches from each of those for the second coin. This gives the outcomes HH, HT, TH, TT.

例如，抛两枚硬币时，树状图的第一层显示第一枚硬币的两种结果（H 或 T），然后从每个结果再分出第二枚硬币的两种结果。这样就得到 HH、HT、TH、TT 四种结果。

Once all outcomes are listed, count the favourable outcomes and the total outcomes to find the probability. For two coins, the probability of getting exactly one head is 2/4 = 1/2.

列出所有结果后，数出有利结果数和总结果数即可求出概率。对于两枚硬币，恰好得到一个正面的概率为 2/4 = 1/2。

11. Common Errors and Exam Tips | 常见错误与考试提示

Many students lose marks in probability questions by making avoidable errors. One common mistake is writing probabilities as ‘1 in 5’ or as ‘20%’ when a fraction is expected. In KS3, probabilities are usually written as simplified fractions unless the question asks for a decimal or percentage.

许多学生在概率题中因可避免的错误而失分。一个常见错误是在要求写分数时把概率写成“1 in 5”或“20%”。在 KS3 阶段，除非题目明确要求用小数或百分数，概率通常应写成最简分数。

Another common mistake is adding probabilities that are not mutually exclusive. Remember that the addition rule P(A or B) = P(A) + P(B) only works when A and B cannot happen at the same time.

另一个常见错误是把并非互斥的事件的概率直接相加。请记住，加法法则 P(A or B) = P(A) + P(B) 只适用于 A 和 B 不能同时发生的情形。

Always write probability as a fraction, decimal, or percentage between 0 and 1.
Simplify fractions fully, such as 3/6 = 1/2.
Check that P(A) + P(not A) = 1.
List all outcomes carefully for combined events.
Do not mix experimental probability with theoretical probability unless the question asks you to compare them.

Published by TutorHao | Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Mastering Basic Probability | 掌握基础概率