📚 Experimental Probability and Relative Frequency | 实验概率与相对频率
In Key Stage 3, probability moves beyond simple chance words to a more numerical approach. You investigate how likely events are by conducting experiments and using relative frequency. This article examines experimental probability, how it differs from theoretical probability, and how practical investigations help us understand randomness and make predictions. We will explore the law of large numbers, the concept of bias, and how to record and analyse data from probability experiments. These skills are essential for tackling questions like those found on page 223 of your Cambridge Checkpoint textbook, where you are asked to design and evaluate experiments.
在 KS3 阶段，概率从简单的“可能”“不可能”等词语转向更量化的方法。你通过实验和相对频率来研究事件发生的可能性。本文探讨实验概率及其与理论概率的区别，以及实际探究如何帮助我们理解随机性并做出预测。我们将学习大数定律、偏差的概念，以及如何记录和分析概率实验数据。这些技能对于解决像剑桥 Checkpoint 教材第 223 页练习 1 这类问题至关重要，那些题目要求你设计并评估实验。
1. What Is Probability? | 什么是概率？
Probability measures how likely an event is to happen. It is expressed as a number between 0 and 1, where 0 means impossible and 1 means certain. In everyday language, we use words like ‘likely’, ‘unlikely’, or ‘even chance’, but mathematics needs precise numbers.
概率衡量事件发生的可能性。它用 0 到 1 之间的数字表示，0 代表不可能，1 代表必然发生。在日常语言中，我们会用“很可能”“不太可能”或“机会均等”这样的词，但数学需要精确的数字。
All probabilities lie on a scale: 0, 0.25, 0.5, 0.75, 1. You can write them as fractions, decimals, or percentages. For example, ‘50% chance’ is a probability of 1/2 or 0.5. Probability can never be negative or greater than 1.
所有概率都落在一个标尺上：0、0.25、0.5、0.75、1。你可以用分数、小数或百分数表示。例如，“50% 的机会”就是概率 1/2 或 0.5。概率永远不会是负数，也不会大于 1。
In KS3, you learn to describe probability using a probability line. Placing events like ‘sun will rise tomorrow’ (near 1) and ‘rolling a 7 on a standard die’ (0) helps build intuition.
在 KS3，你学会用概率线描述事件。将“明天太阳升起”（接近 1）和“掷标准骰子得到 7”（0）这样的事件放在线上，有助于建立直觉。
2. Theoretical Probability | 理论概率
Theoretical probability tells us what we expect to happen when all outcomes are equally likely. If you flip a fair coin, there are two equally likely outcomes: heads or tails. The probability of heads is 1/2. This is found using a simple ratio.
理论概率告诉我们当所有结果等可能时，我们期望发生什么。如果抛一枚公平的硬币，有两种等可能的结果：正面或反面。正面的概率是 1/2。这是通过一个简单的比值得到的。
Probability (event) = Number of favourable outcomes / Total number of equally likely outcomes
理论概率的计算公式为：
概率（事件） = 有利结果数 / 所有等可能结果的总数
For a fair six-sided die, the probability of rolling a 4 is 1/6, because only one face shows 4 out of six possible faces. Similarly, the probability of picking a red ball from a bag containing 3 red and 7 blue balls is 3/10.
对于公平的六面骰子，掷出 4 的概率是 1/6，因为只有一面是 4，而总共有六种可能的面。同样，从装有 3 个红球和 7 个蓝球的袋中取出红球的概率是 3/10。
Theoretical probability assumes perfect randomness and no bias. It provides a solid baseline for comparison when you run actual experiments.
理论概率假设完美的随机性和无偏差。当你进行实际实验时，它提供了一个坚实的比较基准。
3. Introducing Experimental Probability | 实验概率介绍
Experimental probability is derived from the results of an actual experiment. Instead of assuming a spinner is fair, you spin it many times and record the outcomes. The experimental probability of an event is the number of times the event occurs divided by the total number of trials.
实验概率来源于实际实验的结果。你不是假设转盘是公平的，而是旋转它许多次并记录结果。事件的实验概率等于该事件发生的次数除以试验的总次数。
For instance, if you toss a coin 20 times and get heads 13 times, the experimental probability of heads is 13/20 = 0.65. This is quite different from the theoretical 0.5, but that is perfectly normal in a small sample.
例如，如果你抛硬币 20 次，得到 13 次正面，那么正面的实验概率是 13/20 = 0.65。这与理论的 0.5 相差很大，但在小样本中这是完全正常的。
Unlike theoretical probability, experimental values vary each time you repeat the experiment. This variation leads us to the important concept of relative frequency.
与理论概率不同，每次重复实验时实验值都会发生变化。这种变化引出了相对频率的重要概念。
4. Relative Frequency | 相对频率
Relative frequency is simply another term for experimental probability. It tells you the proportion of trials in which an event happened. The formula is straightforward:
相对频率只是实验概率的另一种说法。它告诉你事件发生的试验比例。公式很简单：
Relative frequency = Number of successful trials / Total number of trials
相对频率 = 成功试验次数 / 总试验次数
For example, a spinner lands on blue 8 times in 20 spins. The relative frequency of blue is 8/20 = 0.4, or 40%. This can also be written as a fraction 2/5 or percentage 40%.
例如，一个转盘旋转 20 次，有 8 次停在蓝色区域。蓝色的相对频率是 8/20 = 0.4，即 40%。也可以写作分数 2/5 或百分数 40%。
Relative frequency is always between 0 and 1, just like probability. As you collect more data, you can use it to estimate the true probability of an event, which is especially useful when theoretical probability is unknown.
相对频率始终介于 0 到 1 之间，就像概率一样。随着你收集更多数据，可以用它来估计事件的真实概率，这在理论概率未知时特别有用。
5. Law of Large Numbers | 大数定律
The law of large numbers states that as an experiment is repeated many times, the relative frequency of an event tends to get closer to its theoretical probability. This is a fundamental principle in probability and statistics.
大数定律指出，随着实验重复次数增加，事件发生的相对频率会趋向于其理论概率。这是概率与统计中的一个基本原理。
Imagine flipping a fair coin 10 times – you might get 7 heads, giving a relative frequency of 0.7. If you flip it 100 times, you might get 54 heads (0.54). By the time you reach 1000 flips, the relative frequency is often very close to 0.5, say 0.498.
想象抛一枚公平的硬币 10 次——你可能得到 7 次正面，相对频率为 0.7。如果你抛 100 次，可能得到 54 次正面（0.54）。当你达到 1000 次时，相对频率通常会非常接近 0.5，比如 0.498。
This law does not guarantee that a small set of trials will match theory, but it tells us that more trials give more reliable estimates. That is why scientists and engineers run thousands of tests.
这条定律不保证少量试验会与理论吻合，但它告诉我们，试验次数越多，估计越可靠。这就是为什么科学家和工程师要进行成千上万次测试。
6. Bias and Fairness | 偏差与公平性
An object is called ‘fair’ when all outcomes are equally likely. If an object is biased, some outcomes occur more often than others. Experimental probability helps us investigate whether an object might be biased.
当所有结果等可能时，物体被称为“公平的”。如果物体存在偏差，某些结果会比其它结果更频繁地出现。实验概率可以帮助我们探究物体是否存在偏差。
For example, a weighted die may land on 6 more frequently than 1/6 of the time. By conducting many trials and finding the relative frequency of each outcome, you can compare these with the theoretical probabilities. A large and consistent deviation suggests bias.
例如，一个加重骰子可能以高于 1/6 的频率出现 6 点。通过进行大量试验并找出每个结果的相对频率，你可以将这些结果与理论概率进行比较。持续且较大的偏差暗示存在偏差。
In KS3 exercises such as those on page 223, you are often given data from repeated throws or spins and asked: ‘Do you think the coin is fair? Explain your answer.’ Your reasoning should involve comparing relative frequency to 0.5 and noting the number of trials.
在 KS3 的练习中，比如第 223 页的题目，你经常会得到重复投掷或旋转的数据，并被问到：“你认为这枚硬币公平吗？请解释你的答案。”你的推理应当包含将相对频率与 0.5 进行比较，并注意试验的次数。
7. Recording Results | 记录结果
Organising data carefully is vital when conducting probability experiments. Tally charts and frequency tables allow you to record outcomes as they happen and then summarise them efficiently.
在进行概率实验时，仔细组织数据至关重要。计数符表格和频数表可以让你在结果出现时记录并有效汇总。
A typical frequency table for rolling a die 60 times might look like this:
| Outcome | Tally | Frequency |
|---|---|---|
| 1 | ||| | 3 |
| 2 | || | 2 |
（注：以上为简略示例，实际表格应完整列出所有1–6的结果和频数。）
用中文来说，记录 60 次掷骰子的频数表可以类似建立：列出结果 1 到 6，用画正字计数，最后汇总频数。完成表格后，就可以进入下一步——计算相对频率。
Always give your frequency table a clear title and make sure the total frequency matches the number of trials.
始终给频数表一个清晰的标题，并确保总频数与试验次数一致。
8. Calculating Relative Frequency from Data | 从数据计算相对频率
Once you have a complete frequency table, you can calculate the relative frequency for each outcome. Divide the frequency of the outcome by the total number of trials. Express it as a fraction, a decimal, or a percentage depending on the question.
一旦你有了完整的频数表，就可以计算每个结果的相对频率。用该结果的频数除以总试验次数。根据题目要求，可以表示为分数、小数或百分数。
For the die data above, if ‘4’ appeared 12 times out of 60 rolls, its relative frequency would be 12/60 = 1/5 = 0.2. The theoretical probability is 1/6 ≈ 0.167. The difference is 0.033, which is small, so the die seems fair.
针对上述骰子数据，如果“4”在 60 次中出现 12 次，其相对频率为 12/60 = 1/5 = 0.2。理论概率是 1/6 ≈ 0.167。差值是 0.033，很小，所以骰子看起来是公平的。
When you have many outcomes, you can display relative frequencies in a bar chart or a probability line. This visual representation helps spot unusual patterns or possible bias.
当结果很多时，可以用条形图或概率线显示相对频率。这种视觉表示有助于发现异常模式或可能的偏差。
9. Expected Outcomes | 期望结果
Expected frequency is the number of times you predict an event will occur if the theoretical probability holds. The formula is:
期望频数是指如果理论概率成立，你预测某个事件会发生的次数。公式为：
Expected frequency = Probability of event × Total number of trials
期望频数 = 事件概率 × 试验总次数
For example, if the probability of getting a ‘win’ on a spinner is 1/4 and you spin 200 times, you expect 1/4 × 200 = 50 wins. This does not mean you will get exactly 50, but it is the most likely count over many repetitions.
例如，转盘上出现“赢”的概率是 1/4，你旋转 200 次，那么你期望 1/4 × 200 = 50 次赢。这并不意味着你恰好会得到 50 次，而是多次重复中最可能的次数。
Comparing expected frequency with actual frequency is a key skill. If the actual result is very different from the expected one, you might question whether the probability model is correct or whether the experiment was fair.
将期望频数与实际频数进行比较是一项关键技能。如果实际结果与期望结果差异很大，你就可能质疑概率模型是否正确，或者实验是否公平。
10. Using Data to Test Hypotheses | 用数据检验假设
In experiments, you often start with a hypothesis. A typical one is ‘This coin is fair’, meaning the probability of heads is 0.5. You then carry out, say, 200 flips, calculate the relative frequency, and decide whether the evidence supports fairness.
在实验中，你通常从一个假设开始。一个典型假设是“这枚硬币是公平的”，意思是正面的概率为 0.5。然后你进行 200 次抛掷，计算相对频率，并判断证据是否支持公平性。
If you obtain 120 heads out of 200 flips, the relative frequency is 0.6. With 200 trials, this is quite far from 0.5. That might lead you to suspect bias. However, you must also consider that random variation can produce such a result even with a fair coin; the law of large numbers tells us that even larger trials would be needed to be more certain.
如果你在 200 次抛掷中得到 120 次正面，相对频率是 0.6。在 200 次试验的情况下，这与 0.5 相差甚远，可能会让你怀疑有偏差。然而，你也必须考虑到随机波动也可以在一枚公平硬币上产生这样的结果；大数定律告诉我们，要更有把握还需要更多的试验。
Page 223 exercises often present a table of trial results and ask you to comment on fairness. Always mention the number of trials and how close the relative frequencies are to the expected values.
第 223 页的练习通常会给出一个试验结果表格，要求你评价公平性。记得总要提到试验次数以及相对频率与期望值的接近程度。
11. Common Mistakes | 常见错误
One common mistake is confusing experimental probability with theoretical probability. Students sometimes think that if they get three heads in a row, the next toss must be tails to ‘
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