Experimental Probability and Expected Frequency | 实验概率与期望频率

📚 Experimental Probability and Expected Frequency | 实验概率与期望频率

In KS3 Cambridge Mathematics, probability is not only about calculating a single number from a formula. It also involves carrying out experiments, recording real results and comparing what actually happened with what should happen in theory. This article explains experimental probability, how it differs from theoretical probability, and how to use collected data to make predictions about future outcomes.

在 KS3 剑桥数学中，概率不只是根据公式计算一个数字。它还包括进行实验、记录真实结果，并将实际发生的情况与理论上应发生的情况进行比较。本文将解释实验概率、它与理论概率的区别，以及如何利用收集到的数据预测未来结果。

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1. What Is Experimental Probability? | 什么是实验概率？

Experimental probability is found by doing an experiment or trial and recording what actually happens. It is based on real data rather than on the assumption that all outcomes are equally likely.

实验概率是通过进行实验或试验并记录实际发生的情况而得出的概率。它基于真实数据，而不是基于所有结果等可能的假设。

For example, if you flip a coin 50 times and get heads 28 times, the experimental probability of getting heads is 28 out of 50, which simplifies to 14/25 or 0.56. This means that in your experiment, heads occurred 56% of the time.

例如，如果你抛硬币 50 次，得到正面 28 次，那么出现正面的实验概率是 50 次中的 28 次，化简为 14/25，即 0.56。这意味着在你的实验中，正面出现了 56% 的次数。

• Experimental probability is also called relative frequency.
• 实验概率也称为相对频率。
• It depends on the number of trials: more trials usually give a more reliable estimate.
• 它取决于试验次数：试验次数越多，估计通常越可靠。
• The value can be written as a fraction, decimal or percentage.
• 这个值可以写成分数、小数或百分数。

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2. Key Formula and Notation | 关键公式与符号

The experimental probability of an event is calculated using a simple formula. You divide the number of times the event occurs by the total number of trials.

事件的实验概率可以用一个简单的公式计算。你用事件发生的次数除以试验总次数。

Experimental probability = number of times the event occurs ÷ total number of trials

实验概率 = 事件发生的次数 ÷ 试验总次数

We often use P(event) to mean “the probability of the event”. For experimental data, we can write:

我们常用 P(事件) 表示 “该事件的概率”。对于实验数据，我们可以写成：

P(event) = frequency of the event ÷ total frequency

P(事件) = 事件的频数 ÷ 总频数

For example, if a spinner lands on red 18 times out of 90 spins, then P(red) = 18 ÷ 90 = 1/5 = 0.2 = 20%.

例如，如果一个转盘在 90 次旋转中落在红色区域 18 次，那么 P(红色) = 18 ÷ 90 = 1/5 = 0.2 = 20%。

Always check that all individual frequencies add up to the total frequency before you start. If they do not, your probability will be incorrect.

在开始之前，务必检查所有单个频数相加是否等于总频数。如果不相等，你的概率就会出错。

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3. Theoretical Probability vs Experimental Probability | 理论概率与实验概率

Theoretical probability is what should happen in a perfect world, assuming every outcome is equally likely. Experimental probability is what actually happens when we test it by doing an experiment.

理论概率是在理想情况下应该发生的结果，假设每个结果等可能。实验概率则是我们通过实验实际检验时发生的结果。

For a fair die, the theoretical probability of rolling a 3 is 1/6. If you roll a die 60 times and record a 3 on 8 occasions, the experimental probability is 8/60 = 2/15 ≈ 0.133. This is close to 1/6 ≈ 0.167, but not exactly equal.

对于一枚公平的骰子，掷出 3 的理论概率是 1/6。如果你掷骰子 60 次，记录到 3 出现 8 次，那么实验概率是 8/60 = 2/15 ≈ 0.133。这接近 1/6 ≈ 0.167，但并不完全相等。

Type / 类型	Definition / 定义	Example / 例子
Theoretical probability 理论概率	Based on equally likely outcomes 基于等可能结果	P(heads) = 1/2 P(正面) = 1/2
Experimental probability 实验概率	Based on actual results 基于实际结果	28 heads in 50 flips 50 次抛掷中 28 次正面

The difference between theoretical and experimental probability is called random variation. In a small number of trials, this difference can be quite large. As you increase the number of trials, the experimental probability usually gets closer to the theoretical probability.

理论概率与实验概率之间的差异称为随机波动。在少量试验中，这种差异可能相当大。随着试验次数增加，实验概率通常会越来越接近理论概率。

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4. Expected Frequency | 期望频率

If you know the theoretical probability of an event and the number of trials, you can predict how many times the event should occur. This prediction is called expected frequency.

如果已知某事件的理论概率和试验次数，就可以预测该事件应发生的次数。这个预测值称为期望频率。

Expected frequency = theoretical probability × number of trials

期望频率 = 理论概率 × 试验次数

For example, if a fair die is rolled 300 times, the expected frequency of rolling a 5 is 1/6 ×

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