📚 Investigation 3 - Dice Rolling Experiments | 探究三——掷骰子实验
Rolling dice is one of the most fundamental yet revealing experiments in probability theory. This investigation explores the fairness of dice, the distinction between theoretical and experimental probability, the distribution of sums of two dice, and more advanced concepts such as binomial counts and conditional probability. Through hands-on experiments and data analysis, students can connect abstract mathematics to real-world randomness.
掷骰子是概率论中最基础也最具启发性的实验之一。本探究将探讨骰子的公平性、理论概率与实验概率之间的区别、两骰子之和的分布，以及更高级的概念，如二项计数和条件概率。通过动手实验和数据分析，学生可将抽象的数学与现实世界的随机性联系起来。
1. Introduction to the Investigation | 探究简介
In this investigation, you will conduct a series of dice rolling experiments, record outcomes, and compare them with theoretical predictions. The goal is to understand probability concepts such as relative frequency, expected value, and the law of large numbers.
在本探究中，你将进行一系列掷骰子实验，记录结果，并将其与理论预测进行比较。目标是理解相对频率、期望值和大数定律等概率概念。
A standard six-sided die is assumed fair unless you test for bias. You may use physical dice or computer simulations to collect data.
除非你检验偏差，否则假设标准的六面骰子是公平的。你可以使用实体骰子或计算机模拟来收集数据。
2. Theoretical Probability of a Single Die | 单个骰子的理论概率
{1,2,3,4,5,6}. Each face has probability P(X = x) = 1/6. All outcomes are equally likely.
对于一个公平的六面骰子，样本空间为 S = {1,2,3,4,5,6}。每个面的概率为 P(X = x) = 1/6。所有结果等可能。
mes are 2, 3, 5. These simple calculations form the baseline for experimental comparisons.
掷出偶数的概率为 P(偶数) = 3/6 = 1/2。同理，掷出质数的概率为 3/6，因为质数为 2、3、5。这些简单计算构成了与实验进行比较的基线。
3. Experimental Rolling and Relative Frequency | 实验掷骰与相对频率
To estimate probability experimentally, roll a die many times and record outcomes. The relative frequency of an event A is fA = (number of times A occurs) / (total rolls). As the number of rolls increases, fA tends to approach the theoretical probability – a phenomenon known as the law of large numbers.
要通过实验估计概率，请多次掷骰子并记录结果。事件 A 的相对频率为 fA =（A 发生的次数）/（总掷骰次数）。随着掷骰次数增加，fA 趋于接近理论概率——这一现象称为大数定律。
The table below shows results from 100 simulated rolls of a fair die. Notice how the relative frequencies are close to 1/6 but not exactly equal, reflecting random variation.
下表显示了 100 次模拟掷公平骰子的结果。请注意相对频率如何接近 1/6 但并不完全相等，这反映了随机变异。
| Face | Frequency | Relative Frequency | Theoretical |
|---|---|---|---|
| 1 | 14 | 0.14 | 1/6 ≈ 0.167 |
| 2 | 18 | 0.18 | 1/6 |
| 3 | 17 | 0.17 | 1/6 |
| 4 | 15 | 0.15 | 1/6 |
| 5 | 19 | 0.19 | 1/6 |
| 6 | 17 | 0.17 | 1/6 |
4. Law of Large Numbers in Dice Experiments | 骰子实验中的大数定律
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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