📚 Probability: Sample Space Diagrams and Experimental Probability | 概率：样本空间图与实验概率
Probability is the branch of mathematics that measures how likely an event is to happen. In Key Stage 3, you will use fractions, decimals and percentages to describe probability, draw sample space diagrams for two events, and compare theoretical probability with experimental data.
概率是数学中衡量事件发生可能性大小的一个分支。在 KS3 阶段，你将用分数、小数和百分数来描述概率，为两个事件绘制样本空间图，并将理论概率与实验数据进行比较。
1. The Probability Scale | 概率标尺
All probabilities lie on a scale from 0 to 1. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain. An even chance has probability 0.5, which is the same as 1/2 or 50%.
所有概率都落在 0 到 1 的标尺上。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。对等机会的概率是 0.5，也就是 1/2 或 50%。
You can describe probabilities using words such as impossible, unlikely, even chance, likely and certain. These words must be matched to the correct position on the scale.
你可以用“不可能”“不太可能”“对等机会”“很可能”“一定发生”等词语来描述概率。这些词语必须与标尺上的正确位置相对应。
2. Basic Probability Formula | 基本概率公式
If all outcomes are equally likely, the probability of an event A is given by: P(A) = number of favourable outcomes / total number of possible outcomes.
如果所有结果发生的可能性相等，事件 A 的概率为： P(A) = 有利结果的数量 / 所有可能结果的总数。
For example, when a fair six-sided dice is rolled, the probability of rolling a 5 is 1/6 because there is one favourable outcome and six possible outcomes.
例如，掷一个均匀的六面骰子时，掷出 5 的概率是 1/6，因为有利结果有 1 个，可能结果共有 6 个。
- P(rolling an even number) = 3/6 = 1/2
- P(rolling a number less than 3) = 2/6 = 1/3
- P(rolling a 7) = 0/6 = 0
中文：掷出偶数的概率 = 3/6 = 1/2；掷出小于 3 的数的概率 = 2/6 = 1/3；掷出 7 的概率 = 0/6 = 0。
3. Sample Space Diagrams | 样本空间图
A sample space diagram lists all possible outcomes of two combined events. It helps you count outcomes systematically and find probabilities without missing any possibilities.
样本空间图列出两个组合事件的所有可能结果。它帮助你系统地数出结果，并在不遗漏任何可能性的情况下求概率。
For two spinners each showing 1, 2 or 3, the sample space is shown below:
对于两个分别显示 1、2、3 的转盘，其样本空间如下：
| Spinner B \ Spinner A | 1 | 2 | 3 |
| 1 | (1,1) | (1,2) | (1,3) |
| 2 | (2,1) | (2,2) | (2,3) |
| 3 | (3,1) | (3,2) | (3,3) |
likely outcomes. The probability that the two spinners show the same number is 3/9 = 1/3.
共有 9 个等可能结果。两个转盘转到相同数字的概率是 3/9 = 1/3。
4. Two-Way Tables | 双向表
A two-way table also organises outcomes for two events. For example, a group of 60 students is classified by gender and whether they prefer football or basketball.
双向表也可以整理两个事件的结果。例如，将 60 名学生按性别以及喜欢足球还是篮球进行分类。
| Preference \ Gender | Boys | Girls | Total |
| Football | 18 | 12 | 30 |
| Basketball | 10 | 20 | 30 |
| Total | 28 | 32 | 60 |
om the table, P(a randomly chosen student is a girl who prefers basketball) = 20/60 = 1/3.
从表中可得，随机选择一名学生是喜欢篮球的女生这一概率 = 20/60 = 1/3。
5. Experimental Probability | 实验概率
Experimental probability, also called relative frequency, is found by carrying out a trial or experiment. It is calculated as: Experimental probability = number of successful trials / total number of trials.
实验概率，也叫相对频率，通过进行试验或实验得到。计算方法为： 实验概率 = 成功试验次数 / 试验总次数。
t up 56 times, the experimental probability of landing point up is 56/200 = 28/100 = 0.28.
如果一枚图钉被抛落 200 次，其中 56 次钉尖朝上，则钉尖朝上的实验概率是 56/200 = 28/100 = 0.28。
This value is not fixed: repeating the experiment may give a different result because experimental probability is based on actual outcomes.
这个值不是固定的：重复实验可能会得到不同的结果，因为实验概率基于实际结果。
6. Expected Frequency | 期望频数
You can use theoretical probability to predict how many times an event should occur in a given number of trials. This is the expected frequency: Expected frequency = probability × number of trials.
你可以用理论概率来预测事件在给定的试验次数中应发生的次数。这就是期望频数： 期望频数 = 概率 × 试验次数。
This does not guarantee exactly 250 heads, but it is the average result over many repeats.
例如，抛一枚均匀硬币 500 次，出现正面的期望次数是 0.5 × 500 = 250。这并不保证恰好出现 250 次正面，但它是多次重复后的平均结果。
7. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. For mutually exclusive events A and B: P(A or B) = P(A) + P(B).
如果两个事件不能同时发生，则它们互斥。对于互斥事件 A 和 B： P(A 或 B) = P(A) + P(B)。
g a 2 and rolling a 5 are mutually exclusive. Therefore P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3.
在均匀骰子上，掷出 2 和掷出 5 是互斥事件。因此 P(2 或 5) = 1/6 + 1/6 = 2/6 = 1/3。
Events that can happen together are not mutually exclusive, so you must avoid double-counting any overlap.
能够同时发生的事件不是互斥事件，因此你必须避免重复计算重叠部分。
8. Comparing Theoretical and Experimental Probability | 比较理论概率与实验概率
Theoretical probability uses equally likely outcomes and is calculated from known facts. Experimental probability uses data from an actual experiment. The two may differ, especially for a small number of trials.
理论概率基于等可能结果，并根据已知事实计算。实验概率使用来自实际实验的数据。两者可能不同，尤其是在试验次数较少时。
If a biased coin is tossed 100 times and lands heads 62 times, the experimental probability of heads is 0.62. The theoretical probability for a fair coin would be 0.5, so this suggests the coin may be biased.
如果一枚不均匀的硬币被抛 100 次，其中 62 次正面朝上，则正面的实验概率是 0.62。均匀硬币的理论概率应为 0.5，因此这暗示该硬币可能是不均匀的。
9. Fractions, Decimals and Percentages | 分数、小数与百分数
Probability can be written as a fraction, a decimal or a percentage. You should be able to convert between these forms because exam questions may ask for any one of them.
概率可以写成分数、小数或百分数。你应当能够在这些形式之间转换，因为考试题目可能会要求用其中任意一种形式作答。
- 1/4 = 0.25 = 25%
- 3/5 = 0.6 = 60%
- 1/20 = 0.05 = 5%
中文：1/4 = 0.25 = 25%；3/5 = 0.6 = 60%；1/20 = 0.05 = 5%。
When comparing probabilities, converting them to the same form makes it easier to decide which event is more likely.
在比较概率时，把它们转换成同一种形式会更容易判断哪个事件更可能发生。
10. Exam-Style Worked Example | 考试题型精讲
A bag contains 3 red counters, 2 blue counters and 5 green counters. One counter is chosen at random.
一个袋子里有 3 个红色、2 个蓝色和 5 个绿色筹码。随机抽取一个筹码。
The total number of counters is 3 + 2 + 5 = 10.
筹码总数为 3 + 2 + 5 = 10。
P(red) = 3/10 = 0.3 = 30%
P(not red) = 1 − 3/10 = 7/10 = 0.7 = 70%
he experiment is performed 40 times, the expected number of red counters is 0.3 × 40 = 12.
如果每次抽后放回，并将该实验进行 40 次，则抽到红色筹码的期望次数是 0.3 × 40 = 12。
This example shows how to combine basic probability, complementary events and expected frequency in one question.
这个例子展示了如何在一道题中结合基本概率、互补事件和期望频数。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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