📚 Mastering Probability 5.2: Experimental and Theoretical Probability | 掌握概率 5.2：实验概率与理论概率
Probability is the branch of mathematics that deals with chance and uncertainty. In Cambridge Lower Secondary Mathematics, you need to understand how to calculate theoretical probability, carry out experiments to find experimental probability, and compare the two ideas. This article covers the key concepts in Topic 5.2, including the probability scale, relative frequency, expected frequency, sample space diagrams, and common exam techniques.
概率是数学中研究随机性和不确定性的分支。在剑桥初中数学中，你需要理解如何计算理论概率、通过实验得出实验概率，并比较这两个概念。本文涵盖 5.2 小节的核心内容，包括概率标度、相对频率、期望频数、样本空间图以及常见考试技巧。
1. The Probability Scale | 概率标度
Probability measures how likely an event is to happen. It always takes a value between 0 and 1, where 0 means impossible and 1 means certain. You can write probabilities as fractions, decimals or percentages, but they must never be less than 0 or greater than 1.
概率用来衡量某个事件发生的可能性大小。它的取值始终在 0 到 1 之间，其中 0 表示不可能发生，1 表示一定发生。你可以用分数、小数或百分比来表示概率，但概率绝不能小于 0 或大于 1。
- Impossible / 不可能: 0
- Unlikely / 不太可能: close to 0
- Even chance / 均等机会: 1/2 or 0.5
- Likely / 很可能: close to 1
- Certain / 一定发生: 1
2. Key Terms and Outcomes | 关键术语与结果
An experiment is a repeatable process such as rolling a dice or flipping a coin. An outcome is a single possible result, while an event is a set of one or more outcomes. The sample space is the list of all possible outcomes.
实验是一个可重复的过程，例如掷骰子或抛硬币。结果是一个单独可能发生的情况，而事件则是一个或多个结果的集合。样本空间是所有可能结果的列表。
For example, when you roll a fair six-sided dice, the sample space is {1, 2, 3, 4, 5, 6}. The event "rolling an even number" contains the outcomes {2, 4, 6}.
例如，当你掷一个公平的六面骰子时，样本空间为 {1, 2, 3, 4, 5, 6}。事件“掷出偶数”包含结果 {2, 4, 6}。
Equally likely outcomes mean every outcome has the same chance of occurring. A fair coin has two equally likely outcomes: heads and tails. A biased coin does not have equally likely outcomes.
等可能结果意味着每个结果发生的机会相同。一枚公平的硬币有两种等可能结果：正面和反面。一枚有偏见的硬币则不具有等可能结果。
3. Theoretical Probability | 理论概率
Theoretical probability is calculated using logic and symmetry, without doing an experiment. If all outcomes are equally likely, the probability of an event is given by the formula below.
理论概率是通过逻辑和对称性计算得出的，不需要做实验。如果所有结果等可能，事件发生的概率由以下公式给出。
P(event) = number of favourable outcomes / total number of equally likely outcomes
For a fair six-sided dice, the probability of rolling a 6 is 1/6 because there is one favourable outcome out of six equally likely outcomes. The probability of rolling an even number is 3/6, which simplifies to 1/2.
对于一个公平的六面骰子，掷出 6 的概率为 1/6，因为六个等可能结果中只有一个有利结果。掷出偶数的概率为 3/6，化简后为 1/2。
outcomes in a sample space is always 1. For a dice, 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1.
样本空间中所有互斥结果的概率之和始终为 1。对于骰子，1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1。
4. Experimental Probability | 实验概率
Experimental probability is based on actual trials or observations. It is also called relative frequency. To find it, you divide the number of times an event occurs by the total number of trials.
实验概率基于实际的试验或观察结果。它也被称为相对频率。计算方法是：事件发生的次数除以总试验次数。
Experimental probability = number of successful trials / total number of trials
s 23/50 = 0.46. This is close to the theoretical probability of 0.5 but not exactly equal.
例如，如果你抛硬币 50 次，得到正面 23 次，那么得到正面的实验概率为 23/50 = 0.46。这接近理论概率 0.5，但并不完全相等。
Experimental probability is especially useful when the outcomes are not equally likely, or when it is impossible to list all theoretical outcomes. It gives an estimate based on real data.
当结果不是等可能，或者无法列出所有理论结果时，实验概率尤其有用。它基于真实数据给出一个估计值。
5. Relative Frequency and Estimation | 相对频率与估计
Relative frequency is the same as experimental probability. It helps us estimate the probability of an event when the true probability is unknown. As the number of trials increases, the relative frequency tends to become more stable.
相对频率与实验概率相同。它帮助我们在真实概率未知时估计事件发生的可能性。随着试验次数增加，相对频率往往会变得更加稳定。
This does not mean the experimental probability will exactly equal the theoretical probability every time. It means that with more trials, random variation usually has less effect, so the estimate is more reliable.
这并不意味着实验概率每次都会恰好等于理论概率。它的意思是，随着试验次数增多，随机波动的影响通常会变小，因此估计值更加可靠。
For example, if you toss a coin 10 times you might get 8 heads, giving a relative frequency of 0.8. If you toss it 1000 times, you are much more likely to get a relative frequency close to 0.5.
例如，如果你抛硬币 10 次，可能得到 8 次正面，相对频率为 0.8。如果你抛 1000 次，得到接近 0.5 的相对频率的可能性就大得多。
6. Comparing Theoretical and Experimental Probability | 比较理论概率与实验概率
Theoretical probability tells us what should happen in an ideal world. Experimental probability tells us what actually happened in a particular experiment. The two values are often close but not identical because of random variation.
理论概率告诉我们理想情况下应该发生什么。实验概率告诉我们特定实验中实际发生了什么。由于随机波动，这两个值常常接近但不完全相同。
If an experiment is repeated many times, the experimental probability should get closer to the theoretical probability. This is sometimes called the law of large numbers in informal terms.
如果实验重复多次，实验概率应该会越来越接近理论概率。这有时被非正式地称为大数定律。
If the experimental probability remains very different from the theoretical probability even after many trials, this may suggest that the object is biased or that the assumption of equally likely outcomes is wrong.
如果经过多次试验后，实验概率仍然与理论概率相差很大，这可能表明该物品有偏差，或者“等可能结果”的假设是错误的。
7. Expected Frequency | 期望频数
Expected frequency is the number of times you would expect an event to happen in a given number of trials. It is calculated by multiplying the theoretical probability by the number of trials.
期望频数是在给定试验次数中，你预计某个事件发生的次数。计算方法是用理论概率乘以试验次数。
Expected frequency = probability × number of trials
does not mean you will get exactly 100 sixes, but it is the long-run average expectation.
例如，如果你掷一个公平骰子 600 次，掷出 6 的期望频数为 1/6 × 600 = 100。这并不意味着你一定会恰好得到 100 次 6，但它是长期的平均期望值。
game is 0.2 and 50 people play once each, the expected number of winners is 0.2 × 50 = 10.
期望频数也可用于预测调查或游戏结果。如果赢得游戏的概率为 0.2，50 人各玩一次，那么预期的获胜人数为 0.2 × 50 = 10。
8. Probability of Events Not Happening | 事件不发生的概率
If the probability of an event happening is P(A), then the probability of the event not happening is 1 − P(A). This is because an event either happens or does not happen, and the total probability must be 1.
如果事件 A 发生的概率为 P(A)，那么事件 A 不发生的概率为 1 − P(A)。这是因为一个事件要么发生，要么不发生，总概率必须为 1。
P(not A) = 1 − P(A)
f not rolling a 6 is 1 − 1/6 = 5/6. This is a very useful shortcut in many exam questions.
例如，掷公平骰子得到 6 的概率为 1/6，因此没有得到 6 的概率为 1 − 1/6 = 5/6。这是许多考试题中非常有用的技巧。
Be careful: only use this rule when the two outcomes are complementary. They must cover all possibilities and cannot both happen at the same time.
注意：只有当两个结果互补时才使用这一规则。它们必须覆盖所有可能性，并且不能同时发生。
9. Sample Space Diagrams | 样本空间图
A sample space diagram is a table or list used to show all possible outcomes when two experiments are combined. It helps you count outcomes systematically and avoid missing or repeating any.
样本空间图是用表格或列表来展示两个实验组合时所有可能结果的方法。它帮助你系统地计数结果，避免遗漏或重复。
For example, rolling two fair dice gives 36 equally likely outcomes. A sample space table for the sum of the two scores can be shown as follows.
例如，掷两个公平骰子会产生 36 个等可能结果。两个骰子点数之和的样本空间表如下所示。
| + | 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 |
times out of 36 possible outcomes, so the probability of getting a sum of 7 is 6/36 = 1/6.
从表中可以看出，和为 7 的情况在 36 个可能结果中出现了 6 次，因此得到和为 7 的概率为 6/36 = 1/6。
10. Mutually Exclusive and Combined Events | 互斥事件与组合事件
Mutually exclusive events cannot happen at the same time. For example, when rolling a dice, getting a 2 and getting a 5 are mutually exclusive. If two events A and B are mutually exclusive, you can add their probabilities to find the probability of either happening.
互斥事件不能同时发生。例如，掷骰子时，得到 2 和得到 5 是互斥事件。如果两个事件 A 和 B 互斥，你可以把它们的概率相加，求出其中任一事件发生的概率。
P(A or B) = P(A) + P(B)
ss the events are mutually exclusive, otherwise you may double count overlapping outcomes.
对于公平骰子，P(2 或 5) = 1/6 + 1/6 = 2/6 = 1/3。除非事件互斥，否则不要简单相加概率，否则你可能会重复计算重叠的结果。
11. Bias and Fairness | 偏差与公平性
An object is fair if all possible outcomes are equally likely. A fair coin has probability 1/2 of landing heads and 1/2 of landing tails. A biased object has outcomes with unequal probabilities.
如果所有可能结果等可能，则该物品是公平的。公平硬币正面朝上的概率为 1/2，反面朝上的概率为 1/2。有偏差的物品其结果概率不相等。
You can investigate bias by carrying out an experiment and comparing the experimental probability with the theoretical probability. If a coin is tossed many times and gives heads far more often than 0.5, it may be biased.
你可以通过实验并比较实验概率与理论概率来检验偏差。如果一枚硬币抛了很多次后，正面出现的频率远高于 0.5，那么它可能是有偏差的。
However, you must be cautious: a small number of trials can give a misleading result just by chance. Always look at the number of trials before making a conclusion about bias.
但是，你必须谨慎：少量试验可能会因偶然性而给出误导性结果。在得出关于偏差的结论之前，一定要关注试验次数。
12. Common Misconceptions and Exam Tips | 常见误区与考试技巧
One common mistake is adding probabilities for events that are not mutually exclusive. Another is forgetting that probabilities must be between 0 and 1. Always simplify fractions when possible unless the question asks for a decimal or percentage.
一个常见错误是把非互斥事件的概率相加。另一个是忘记概率必须在 0 到 1 之间。除非题目要求用小数或百分比表示，否则应尽量化简分数。
- Always show your working clearly / 始终清楚展示计算过程.
- Check that your probability is not greater than 1 or less than 0 / 检查概率是否在 0 到 1 之间.
- Use the complement rule to save time, such as P(not A) = 1 − P(A) / 使用补集规则节省时间，如 P(非 A) = 1 − P(A).
- For combined events, use a sample space diagram or list all outcomes / 对于组合事件，使用样本空间图或列出所有结果.
- Compare experimental and theoretical probability only when the number of trials is large / 仅在试验次数较多时比较实验概率与理论概率.
By mastering these skills, you will be confident with Cambridge KS3 probability questions involving theoretical probability, experimental probability, sample spaces and expected frequency.
掌握这些技能后，你就能自信地应对剑桥 KS3 概率题，包括理论概率、实验概率、样本空间和期望频数等问题。
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