📚 Experimental Probability (p.224) | 实验概率（第224页）
Probability is everywhere – from weather forecasts to the chance of winning a board game. In KS3 Cambridge Mathematics, you learn about both theoretical and experimental probability. This article, based on page 224, explores how we can estimate probability by carrying out experiments and calculating relative frequency.
概率无处不在——从天气预报到赢得棋盘游戏的可能性。在 KS3 剑桥数学课程中，你要学习理论概率和实验概率。这篇基于教材第 224 页的文章将探索如何通过进行实验并计算相对频率来估计概率。
1. What is Probability? | 什么是概率？
Probability is a number between 0 and 1 that tells us how likely an event is to happen. A probability of 0 means the event is impossible, and a probability of 1 means it is certain. Probabilities can be written as fractions, decimals or percentages.
概率是一个介于 0 到 1 之间的数字，用来表示某个事件发生的可能性大小。概率为 0 表示该事件不可能发生，概率为 1 表示该事件必然发生。概率可以用分数、小数或百分数来表示。
2. Theoretical Probability | 理论概率
Theoretical probability is worked out using reasoning and assumes all outcomes are equally likely. The formula is: P(Event) = number of favourable outcomes ÷ total number of possible outcomes. For example, when you flip a fair coin, P(Head) = 1 ÷ 2 = 0.5.
理论概率是通过推理得出的，它假设所有结果出现的可能性相等。计算公式为：P(事件) = 有利结果的数量 ÷ 所有可能结果的总数。例如，抛一枚均匀的硬币时，出现正面的概率 P(正面) = 1 ÷ 2 = 0.5。
3. Introduction to Experimental Probability | 实验概率简介
Experimental probability is found by actually carrying out trials or experiments. Instead of assuming equal likelihood, we look at what happened when we repeated the event many times. This gives us an estimate of the true probability based on real data.
实验概率是通过实际进行试验或实验得到的。我们不假设所有结果等可能，而是观察多次重复事件时实际发生了什么。这就给了我们一个基于真实数据的真实概率的估计值。
4. Calculating Relative Frequency | 计算相对频率
Relative frequency is another name for experimental probability. It is calculated using: Relative frequency = number of times the event occurs ÷ total number of trials. As the number of trials increases, the relative frequency usually gets closer to the theoretical probability.
相对频率是实验概率的另一个名称。它的计算公式是：相对频率 = 事件发生的次数 ÷ 试验总次数。随着试验次数的增加，相对频率通常会越来越接近理论概率。
5. Conducting a Simple Experiment | 进行一个简单实验
Let’s imagine we want to find the experimental probability of getting heads when flipping a coin. We flip the coin 50 times and record the result each time. At the end, we count how many heads appeared and divide by 50. If we got 26 heads, the relative frequency is 26/50 = 0.52.
想象一下，我们想要求出抛硬币得到正面的实验概率。我们将硬币抛 50 次，并记录每次的结果。最后，数一数正面出现了多少次，再除以 50。如果得到了 26 次正面，那么相对频率就是 26/50 = 0.52。
6. Comparing Theoretical and Experimental Results | 比较理论与实验结果
In our coin experiment, the theoretical probability was 0.5, but the experimental probability from 50 trials was 0.52. These values are close but not identical. The small difference is due to random variation. We do not expect the experiment to match the theory perfectly every time.
在我们的硬币实验中，理论概率是 0.5，但 50 次试验得到的实验概率是 0.52。这两个值很接近但并不相同。这个微小的差异是由随机波动造成的。我们不可能期望每次实验都完美地符合理论值。
7. The Law of Large Numbers | 大数定律
The law of large numbers states that as you repeat an experiment more and more times, the relative frequency tends to settle down towards the theoretical probability. If we flipped the coin 1000 times, the proportion of heads would be very close to 0.5.
大数定律指出，当你将一个实验重复很多很多次时，相对频率会趋于稳定，并接近理论概率。如果我们抛硬币 1000 次，正面的比例就会非常接近 0.5。
8. Worked Example: Coin Toss | 实例分析：抛硬币
Here is a record of 50 coin tosses. The table shows the outcomes and frequencies.
下表是 50 次抛硬币的记录，显示了结果和频数。
| Outcome (结果) | Frequency (频数) | Relative Frequency (相对频率) |
|---|---|---|
| Heads (正面) | 26 | 26/50 = 0.52 |
| Tails (反面) | 24 | 24/50 = 0.48 |
Notice that the two relative frequencies add up to 1 (0.52 + 0.48 = 1), which must be true because heads and tails cover all possible outcomes.
注意到两个相对频率之和为 1 (0.52 + 0.48 = 1)，这必然成立，因为正面和反面涵盖了所有的可能结果。
9. Worked Example: Rolling a Die | 实例分析：掷骰子
Suppose a six-sided die is rolled 60 times and the number 4 appears 9 times. The theoretical probability of rolling a 4 is 1/6 ≈ 0.167. The experimental probability is 9/60 = 0.15. Again, they are close. If the die were biased, we might see a much larger difference.
假设一个六面骰子被掷了 60 次，其中数字 4 出现了 9 次。掷出 4 的理论概率是 1/6 ≈ 0.167。实验概率是 9/60 = 0.15。这两个值再次很接近。如果骰子存在偏差，我们可能会看到大得多的差异。
10. Bias and Fairness | 偏差与公平性
When the experimental probability stays consistently far from the theoretical probability even after many trials, we might suspect that the object is biased. For example, a coin that gives 70 heads out of 100 tosses may be uneven in weight. Experimental data helps us test whether a game or object is fair.
如果即使经过很多次试验，实验概率仍然一直远离理论概率，我们就会怀疑这个物体存在偏差。例如，一枚硬币在 100 次投掷中出现 70 次正面，那么它的重量可能不均匀。实验数据帮助我们检验某个游戏或物体是否公平。
11. Using Relative Frequency for Predictions | 利用相对频率进行预测
Once we have a reliable relative frequency, we can use it to make predictions. If a spinner lands on blue 40 times out of 120 spins, the relative frequency is 40/120 = 1/3. We can then predict that in 300 spins, blue would appear about 100 times (300 × 1/3 = 100).
一旦我们得到了一个可靠的相对频率，就可以用它来进行预测。如果一个转盘在 120 次转动中有 40 次停在蓝色区域，那么相对频率就是 40/120 = 1/3。我们就可以预测，在 300 次转动中，蓝色将出现约 100 次（300 × 1/3 = 100）。
12. Summary & Key Points | 总结与要点
Theoretical probability is based on equally likely outcomes, while experimental probability comes from real trials. Relative frequency equals occurrences divided by total trials. The law of large numbers tells us that more trials give us a value closer to the theoretical probability. These ideas are crucial in statistics and everyday decision-making.
理论概率基于等可能的结果，而实验概率来自真实的试验。相对频率等于事件发生次数除以试验总次数。大数定律告诉我们，试验次数越多，得到的值就越接近理论概率。这些概念在统计学和日常决策中至关重要。
- Probability scale: 0 (impossible) to 1 (certain) / 概率范围：0（不可能）到 1（一定发生）
- Theoretical P = favourable outcomes / total outcomes / 理论概率 = 有利结果数 ÷ 总结果数
- Relative frequency = occurrences / trials / 相对频率 = 发生次数 ÷ 试验次数
- Increased trials → value approaches theoretical probability / 增加试验 → 数值趋近理论概率
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