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

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

Probability is the branch of mathematics that measures the likelihood of an event happening. It ranges from 0 (impossible) to 1 (certain), and we can express it as a fraction, decimal, or percentage. In this article, we will explore the difference between experimental probability, which is based on actual trials, and theoretical probability, which is calculated by reasoning about equally likely outcomes. You will learn how to record data, calculate probabilities, and understand why results can vary between experiment and theory.

概率是衡量某一事件发生可能性的数学分支，其值范围从 0（不可能）到 1（一定发生），可以用分数、小数或百分比表示。本文将探讨实验概率（基于实际试验）与理论概率（通过等可能结果推理计算得出）之间的区别。你将学习如何记录数据、计算概率，并理解为什么实验结果与理论值可能存在差异。

1. What is Probability? | 什么是概率？

Probability is a measure of how likely something is to happen. It is always a number between 0 and 1. An event with a probability of 0 will never happen, while an event with a probability of 1 is certain to happen. The probability of any event A can be written as P(A). For example, if we toss a fair coin, there are two possible outcomes: heads or tails. Since both are equally likely, the theoretical probability of getting heads is 1/2.

概率是衡量某件事发生可能性的度量。它始终是介于 0 和 1 之间的一个数。概率为 0 的事件永远不会发生，概率为 1 的事件肯定发生。任何事件 A 的概率可以记作 P(A)。例如，抛一枚公平硬币时，有两种可能的结果：正面或反面。由于两者可能性相同，得到正面的理论概率为 1/2。

We can show the probability scale on a number line from 0 to 1. Words like ‘impossible’, ‘unlikely’, ‘even chance’, ‘likely’, and ‘certain’ help describe probabilities without using numbers.

我们可以用一条从 0 到 1 的数轴来表示概率标尺。像“不可能”、“不大可能”、“对半机会”、“很可能”和“肯定”这样的词语有助于在不用数字的情况下描述概率。

Probability in words	Numerical value
Impossible	0
Even chance	0.5
Certain	1

2. Theoretical Probability | 理论概率

Theoretical probability is calculated without doing an experiment. It assumes that all outcomes are equally likely. The formula is:

理论概率不需要通过实验来计算，它假设所有结果出现的可能性相等。计算公式为：

P(event) = Number of favourable outcomes ÷ Total number of possible outcomes

For instance, when rolling a fair six-sided die, the probability of rolling a 4 is 1/6 because there is one favourable face ‘4’ and six possible outcomes in total. Similarly, the probability of rolling an even number is 3/6, which simplifies to 1/2, because the even outcomes are 2, 4, and 6.

例如，掷一枚均匀的六面骰子时，掷出 4 的概率是 1/6，因为有利的面（4）只有一个，总共可能的结果有六个。同样，掷出偶数的概率是 3/6，化简为 1/2，因为偶数结果包括 2、4 和 6。

Theoretical probability gives an exact prediction based on symmetry. It is often used in designing fair games and in weather forecasting models, but it always requires that outcomes are truly equally likely.

理论概率基于对称性给出精确的预测。它常用于设计公平游戏以及天气预报模型中，但始终要求各个结果确实是等可能的。

3. Experimental Probability | 实验概率

Experimental probability is found by carrying out a practical investigation and recording results. The formula is:

实验概率通过实际进行调查并记录结果来得出。计算公式为：

Experimental probability = Number of times the event occurred ÷ Total number of trials

If we toss a drawing pin (thumbtack) 50 times and it lands point-up 38 times, the experimental probability of landing point-up is 38/50 = 0.76. Notice that experimental probability varies from one set of trials to another, especially when the number of trials is small.

如果我们抛掷一枚图钉 50 次，有 38 次针尖朝上落地，那么针尖朝上的实验概率为 38/50 = 0.76。要注意，实验概率会随不同批次的试验而变化，尤其是当试验次数较少时。

Experimental probability is also called relative frequency. It becomes more stable and closer to the theoretical probability as you increase the number of trials. This is known as the law of large numbers.

实验概率也称作相对频率。随着试验次数的增加，它会变得更加稳定，更接近理论概率。这就是大数定律。

4. Collecting Data and Relative Frequency | 收集数据与相对频率

When running an experiment, you should use a tally chart or a frequency table to record outcomes clearly. After each set of trials, the relative frequency is calculated. Let us imagine spinning a spinner with colours red, blue, green, and yellow 100 times. The results are:

进行实验时，应当使用划记表或频数表清晰地记录结果。每一批试验结束后，计算相对频率。设想一个带有红、蓝、绿、黄四种颜色的转盘被转动 100 次，结果如下：

Red: 24 times
Blue: 31 times
Green: 19 times
Yellow: 26 times

The relative frequency of red is 24/100 = 0.24 or 24%. Even though the theoretical probability might be 0.25 for each colour if the spinner is fair, the experimental results show small differences. This variation is normal.

红色的相对频率是 24/100 = 0.24 即 24%。如果转盘是均匀的，每个颜色的理论概率可能都是 0.25，但实验结果显示出微小的差异，这种偏差是正常的。

5. Comparing Experimental and Theoretical Probability | 比较实验概率与理论概率

To compare, we look at the difference between the experimental relative frequency and the theoretical probability. For the spinner results above, the difference for red is |0.24 – 0.25| = 0.01. As we perform more spins, this difference should reduce. In many KS3 investigations, you will be asked to draw a bar chart or a line graph showing relative frequency over trials.

比较时，我们计算实验相对频率与理论概率之间的差值。上面转盘数据中，红色的差值为 |0.24 – 0.25| = 0.01。随着转动次数的增加，这个差值应该会缩小。在许多 KS3 实验中，你都会被要求绘制显示不同试验次数下相对频率的条形图或折线图。

If the experimental probability is very different from the theoretical probability, you should question whether the sample size is sufficient or whether the assumptions of fairness are correct. For example, a biased coin would show a long-run experimental probability noticeably different from 0.5.

如果实验概率与理论概率相差很大，就应该思考样本量是否足够，或者公平性的假设是否正确。例如，一枚有偏倚的硬币在长期实验中会显示出与 0.5 明显不同的实验概率。

6. Fairness and Bias | 公平性与偏倚

A fair experiment is one where every outcome has an equal chance of occurring. A six-sided die is fair if each number from 1 to 6 has a probability of 1/6. If a die is loaded, some numbers will appear more often than others. We can test for bias by conducting a large number of trials and comparing the experimental probability to the expected theoretical value.

公平实验是指每个结果发生的可能性相等。如果一枚六面骰子是均匀的，那么 1 到 6 每个数字出现的概率都是 1/6。如果骰子被做过手脚，某些数字出现的频率就会高于其他数字。我们可以通过大量试验并比较实验概率与预期的理论值来检验是否存在偏倚。

For instance, suppose we roll a suspicious die 600 times. We expect each number about 100 times. If the number 3 appears only 70 times while number 5 appears 130 times, we might suspect bias. The experimental probability for number 5 would be 130/600 ≈ 0.217, giving a difference of roughly 0.05 from the theoretical 0.1667.

例如，假设我们投掷一枚可疑的骰子 600 次，预期每个数字大约出现 100 次。如果数字 3 只出现 70 次，而数字 5 出现了 130 次，就应当怀疑存在偏倚。数字 5 的实验概率为 130/600 ≈ 0.217，与理论值 0.1667 相差了约 0.05。

7. The Law of Large Numbers | 大数定律

The law of large numbers states that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability. This is why scientists and engineers repeat experiments many times to get reliable results. In a classroom setting, you might toss a coin only 10 times and get 7 heads, giving an experimental probability of 0.7. But after 1000 tosses, the relative frequency of heads will be very close to 0.5.

大数定律指出，随着试验次数的增加，实验概率会趋向于理论概率。这就是科学家和工程师反复进行实验以获得可靠结果的原因。在课堂上，如果你只抛 10 次硬币，可能得到 7 次正面，实验概率为 0.7；但抛 1000 次后，正面的相对频率将会非常接近 0.5。

The law does not guarantee exact equality at any finite number of trials, but it explains why random events exhibit a predictable pattern in the long run. This concept is fundamental in statistics and helps us understand sample sizes in surveys.

该定律并不保证在任何有限次试验中实验概率与理论概率完全相等，但它解释了为什么随机事件在长期中会呈现出可预测的模式。这一概念是统计学的基础，有助于我们理解调查中的样本量问题。

8. Sample Space and Equally Likely Outcomes | 样本空间与等可能结果

The sample space is the set of all possible outcomes of an experiment. For tossing two fair coins, the sample space is {HH, HT, TH, TT}. If the coins are fair, each of these four outcomes is equally likely, so the probability of getting two heads is 1/4. Listing the sample space systematically helps you find theoretical probabilities accurately.

样本空间是指一个实验所有可能结果的集合。抛两枚公平硬币时，样本空间为 {HH, HT, TH, TT}。若硬币是公平的，这四个结果中的每一个都等可能，所以得到两个正面的概率是 1/4。系统地列出样本空间有助于准确求出理论概率。

When the number of outcomes is larger, a two-way table or a tree diagram can be used to list all possibilities. In KS3 Mathematics, you are expected to use these tools to solve problems involving combined events.

当结果数量较多时，可以使用双向表或树状图列出所有可能性。在 KS3 数学中，你需要运用这些工具来解决涉及组合事件的问题。

9. Expressing Probability in Different Forms | 用不同形式表示概率

Probability can be written as a fraction, a decimal, or a percentage. For example, a theoretical probability of 1/4 can also be written as 0.25 or 25%. In experimental work, it is common to present relative frequency as a decimal rounded to an appropriate number of significant figures. Changing between forms is an essential skill. To convert a fraction to a decimal, divide the numerator by the denominator. To change to a percentage, multiply the decimal by 100.

概率可以用分数、小数或百分比表示。例如，理论概率 1/4 也可以写作 0.25 或 25%。在实验工作中，相对频率通常表示为保留适当有效数字的小数。在不同形式之间进行转换是一项必备技能。将分数转换为小数时，用分子除以分母；转换为百分比时，再将小数乘以 100。

When comparing probabilities, converting them to the same form often makes it easier to decide which event is more likely. For example, which is greater: 0.3 or 33%? Converting 33% to 0.33 shows that 33% is slightly larger.

在比较概率时，将它们转化为相同的形式通常能更容易地判断哪个事件更可能发生。例如，0.3 和 33% 哪个更大？把 33% 转换为 0.33 后可知，33% 稍大一些。

10. Designing a Probability Experiment | 设计概率实验

When designing an experiment for KS3, you should first identify the aim and make a prediction based on theoretical probability. Then, decide on the number of trials and the method of recording data. For example, to test if a drawing pin is biased, you could drop it 200 times and record the number of ‘point up’ and ‘point sideways’ outcomes.

在为 KS3 设计实验时，你应当先明确实验目的，并根据理论概率做出预测。然后决定试验次数和数据记录方法。例如，为了检验一枚图钉是否有偏倚，可以让图钉掉落 200 次，记录“针尖朝上”和“针尖侧卧”两种结果的次数。

Always repeat trials under the same conditions. Collect data in a table, compute relative frequencies, and plot a graph if required. At the end, compare your experimental probability with your earlier theoretical prediction. Discuss any differences and whether you think there is evidence of bias.

始终在相同条件下重复试验。用表格收集数据，计算相对频率，如有需要则绘制图表。最后，将实验概率与之前的理论预测作比较，讨论差异并说明是否认为存在偏倚的证据。

11. Common Misconceptions and Tips | 常见误区与提示

One common mistake is to believe that if you flip a coin and get heads five times in a row, the next flip is more likely to be tails. This is incorrect; the coin has no memory, and each flip remains independent with a probability of 1/2 for heads. Another error is confusing experimental probability with theoretical probability for a small number of trials.

一个常见的错误是认为如果连续五次抛硬币都得到正面，下一次抛掷出现反面的可能性会更大。这是错误的：硬币没有记忆能力，每次抛掷仍然是独立的，得到正面的概率始终是 1/2。另一个错误是混淆小样本试验中的实验概率与理论概率。

Always remember that results from a small number of trials can be misleading. Use a large sample to draw reliable conclusions. When calculating probability, check that you have identified the sample space correctly and that all outcomes are truly equally likely.

请始终牢记，少量试验得出的结果可能具有误导性。要得出可靠结论，应使用大样本。计算概率时，要确认你已经正确找出了样本空间，并且所有结果确实是等可能的。

12. Practice Questions | 练习题

Here are some typical KS3 questions to check your understanding:

以下是一些典型的 KS3 题目，用来检验你的理解：

A fair 8-sided spinner numbered 1 to 8 is spun. What is the theoretical probability of landing on a prime number? (Answer: Primes are 2, 3, 5, 7, so 4/8 = 1/2)

一个标有数字 1 到 8 的均匀 8 边转盘被转动一次。转到质数的理论概率是多少？
In 300 rolls of a die, the number 6 appeared 56 times. Calculate the experimental probability of rolling a 6 as a fraction and as a percentage. (Answer: 56/300 = 14/75, approximately 18.7%)

掷一枚骰子 300 次，数字 6 出现了 56 次。计算掷出 6 的实验概率，并分别用分数和百分比表示。
A bag contains 5 red, 3 blue, and 2 green counters. One counter is taken at random. What is the probability it is not red? (Answer: total counters = 10, not red = 5, probability = 5/10 = 1/2)

一个袋子里装有 5 个红色、3 个蓝色和 2 个绿色筹码。随机取出一个，它不是红色的概率是多少？

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