Understanding Probability | 理解概率

📚 Understanding Probability | 理解概率

In KS3 Cambridge mathematics, probability is the branch of mathematics that describes how likely an event is to happen. You will meet probability in everyday situations, from weather forecasts to simple games. This unit introduces the probability scale, basic notation, and the difference between experimental and theoretical probability. It also prepares you for harder topics such as sample space diagrams and combined events in later stages.

在 KS3 剑桥数学中，概率是描述一个事件发生可能性的数学分支。你会在日常生活中遇到概率，从天气预报到简单游戏。本单元介绍概率标度、基本符号，以及实验概率与理论概率的区别。它也为你后续学习样本空间图和组合事件等更难的内容打下基础。

1. What is Probability? | 什么是概率？

Probability is a measure of how likely an event is to occur. It always takes a value from 0 to 1. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain.

概率是衡量一个事件发生可能性的度量。它的取值总是在 0 到 1 之间。概率为 0 表示事件不可能发生，概率为 1 表示事件一定发生。

We use the word ‘event’ to describe any outcome or set of outcomes from a trial or experiment. For example, rolling a 6 on a fair die is an event.

我们用事件一词来描述试验或实验中任意一个结果或一组结果。例如，掷一枚公平骰子得到 6 就是一个事件。

2. The Probability Scale | 概率标度

The probability scale is a number line from 0 to 1. We can place words such as impossible, unlikely, even chance, likely and certain along the scale.

概率标度是一条从 0 到 1 的数轴。我们可以把不可能、不太可能、机会均等、很可能和一定等词语放在标度上。

Event	Probability	Description
Impossible	0	It cannot happen
Unlikely	close to 0	It probably will not happen
Even chance	0.5	It is equally likely to happen or not happen
Likely	close to 1	It probably will happen
Certain	1	It will definitely happen

You should be able to place the probability of any event on this scale. For example, the probability that a fair coin lands heads is 0.5, so it sits exactly in the middle.

你应该能够把任意事件的概率对应到这条标度上。例如，公平硬币正面朝上的概率是 0.5，因此它正好位于中间。

3. Basic Probability Notation | 基本概率符号

We often write P(A) to mean the probability of event A. For example, P(head) means the probability of getting a head when you flip a fair coin.

我们常用 P(A) 表示事件 A 的概率。例如，P(正面) 表示掷一枚公平硬币得到正面的概率。

The formula for theoretical probability is:

P(A) = number of favourable outcomes / total number of outcomes

理论概率的公式如下：

P(A) = 有利结果数 / 总结果数

The outcomes must be equally likely for this formula to work. For example, all faces of a fair die are equally likely.

这个公式只有在所有结果等可能时才成立。例如，公平骰子的每一个面出现的机会都相同。

4. Experimental vs Theoretical Probability | 实验概率与理论概率

Theoretical probability is what we expect to happen based on equally likely outcomes. Experimental probability is what actually happens in an experiment or trial.

理论概率是我们根据等可能结果预期发生的情况。实验概率是在实验或试验中实际发生的情况。

If you flip a coin 100 times and get 45 heads, the experimental probability of heads is 45/100 = 0.45, which is close to the theoretical value 0.5.

如果你掷硬币 100 次得到 45 次正面，那么正面的实验概率为 45/100 = 0.45，接近理论值 0.5。

A small number of trials can give an experimental probability that is very different from the theoretical probability. That is why we usually need many trials before making a conclusion.

少量试验得到的实验概率可能与理论概率差别很大。因此我们通常需要很多次试验才能得出结论。

5. Calculating Simple Probabilities | 计算简单概率

To calculate a simple probability, count the favourable outcomes and divide by the total number of possible outcomes. For a fair six-sided die, the probability of rolling an even number is 3/6 = 1/2.

要计算简单概率，先数出有利结果数，再除以所有可能结果的总数。对于公平六面骰子，掷出偶数的概率是 3/6 = 1/2。

You can simplify fractions when possible. For example, the probability of selecting a red card from a standard 52-card deck is 26/52 = 1/2.

如果可能，你可以对分数进行化简。例如，从标准 52 张扑克牌中抽到红牌的概率是 26/52 = 1/2。

Always start by listing all possible outcomes if you are not sure. This helps you avoid missing or double-counting outcomes.

如果不确定，总是先列出所有可能结果。这能帮助你避免漏算或重复计数结果。

6. Probability of an Event Not Happening | 事件不发生的概率

The probability of an event A not happening is written P(not A). Because the total probability of all possible outcomes is 1, we have:

事件 A 不发生的概率记作 P(非 A)。因为所有可能结果的概率总和为 1，所以我们有：

P(not A) = 1 − P(A)

P(非 A) = 1 − P(A)

For example, if the probability of rain tomorrow is 0.3, then 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，那么不下雨的概率就是 1 − 0.3 = 0.7。

This rule is very useful because sometimes it is easier to find the probability of the opposite event first.

这个规则非常有用，因为有时先求相反事件的概率会更容易。

7. Sample Space Diagrams | 样本空间图

A sample space is the set of all possible outcomes of an experiment. We can list outcomes or use a table. For two coins flipped together, the sample space is HH, HT, TH, TT.

样本空间是一个实验所有可能结果的集合。我们可以列出结果或使用表格。同时掷两枚硬币时，样本空间是 HH、HT、TH、TT。

A sample space diagram can help you count outcomes systematically. The probability of getting two heads is 1/4 because there are 4 equally likely outcomes and only 1 favourable outcome.

样本空间图可以帮助你系统地计数结果。抛出两个正面的概率是 1/4，因为有 4 个等可能结果，而有利结果只有 1 个。

For two six-sided dice, you can draw a 6 by 6 table. There are 36 equally likely outcomes, and the sum of the two dice can range from 2 to 12.

对于两个六面骰子，你可以画一个 6 乘 6 的表格。共有 36 个等可能结果，两个骰子点数之和的范围是 2 到 12。

8. Mutually Exclusive Events | 互斥事件

Two events are mutually exclusive if they cannot happen at the same time. For example, rolling a 2 and rolling a 5 on a single die are mutually exclusive events.

如果两个事件不能同时发生，则它们互斥。例如，掷一个骰子，掷出 2 和掷出 5 是互斥事件。

If A and B are mutually exclusive, then P(A or B) = P(A) + P(B). This is called the addition rule.

如果 A 和 B 互斥，那么 P(A 或 B) = P(A) + P(B)。这称为加法法则。

Example: the probability of rolling a 2 or a 5 on a fair die is 1/6 + 1/6 = 2/6 = 1/3.

例如，掷一枚公平骰子得到 2 或 5 的概率是 1/6 + 1/6 = 2/6 = 1/3。

Mutually exclusive events are often tested with spinners, dice and cards. Always check whether both events can happen together before using the addition rule.

互斥事件经常通过转盘、骰子和扑克牌来考查。在使用加法法则之前，一定要检查两个事件是否可能同时发生。

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

Relative frequency is the number of times an event occurs divided by the total number of trials. It is used to estimate probability from experiments.

相对频率是事件发生的次数除以试验总次数。它用于通过实验估计概率。

For example, if a football player scores 18 goals from 60 shots, the relative frequency of scoring is 18/60 = 0.3. We can say the estimated probability of scoring is 0.3.

例如，一名足球运动员射门 60 次打进 18 球，进球相对频率为 18/60 = 0.3。我们可以说进球概率的估计值为 0.3。

More trials usually give a relative frequency closer to the true theoretical probability. This is called the law of large numbers.

试验次数越多，相对频率通常越接近真实的理论概率。这称为大数定律。

If a spinner is biased, you can only estimate its probabilities by collecting data from many spins and finding the relative frequencies.

如果转盘有偏，你只能通过多次旋转收集数据并求出相对频率来估计其概率。

10. Common Misconceptions | 常见误区

A common mistake is to think that if a coin has landed heads five times in a row, it is more likely to land tails next. In fact, each flip is independent, and the probability remains 1/2 for tails.

一个常见错误是认为硬币连续五次正面朝上后，下一次更可能反面朝上。实际上，每次抛掷都是独立的，反面的概率仍然是 1/2。

Another misconception is adding probabilities for non-mutually exclusive events. For events that can overlap, you must subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).

另一个误区是对非互斥事件直接相加概率。对于可能重叠的事件，你必须减去重叠部分：P(A 或 B) = P(A) + P(B) − P(A 与 B)。

Always check whether outcomes are equally likely before using the basic formula. If a die is biased, each face may not have probability 1/6.

在使用基本公式之前，始终检查结果是否等可能。如果骰子有偏，每个面出现的概率可能不是 1/6。

Remember that probability does not tell you exactly what will happen on one single trial. It only describes how likely an outcome is in the long run.

记住，概率并不能告诉你单次试验一定会发生什么。它只描述某结果在长期内发生的可能性。

Published by TutorHao | Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Understanding Probability | 理解概率