Experimental Probability in IB Mathematics | IB数学：实验概率的理解与应用

📚 Experimental Probability in IB Mathematics | IB数学：实验概率的理解与应用

Experimental probability is a core concept in the IB Mathematics curriculum. It is based on actual trials or simulations rather than theoretical assumptions, and it forms a bridge between abstract probability theory and real-world observation.

实验概率是IB数学课程中的一个核心概念。它基于实际试验或模拟，而非理论假设，是连接抽象概率论与现实观察之间的桥梁。

1. Definition and Fundamental Idea | 定义与基本思想

Experimental probability is defined as the ratio of the number of times an event occurs to the total number of trials performed. It is also called empirical probability or relative frequency.

实验概率的定义是事件发生的次数与总试验次数之比。它也被称为经验概率或相对频率。

P(event) ≈ number of favourable outcomes ÷ total number of trials

P(事件) ≈ 有利结果数 ÷ 总试验次数

For example, if a coin is tossed 100 times and lands heads 47 times, the experimental probability of heads is 47/100 = 0.47.

例如，一枚硬币抛掷100次，正面出现47次，则正面的实验概率为47/100 = 0.47。

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

Theoretical probability is calculated using known mathematical structures, assuming equally likely outcomes. Experimental probability is derived from observed data, so it may differ from the theoretical value, especially when the sample size is small.

理论概率利用已知的数学结构计算，假设各结果等可能。实验概率则来自观测数据，因此可能不同于理论值，尤其是在样本量较小时。

Feature	Theoretical Probability	Experimental Probability
Basis	Known outcomes and symmetry	Repeated trials or simulations
Value	Fixed for a given model	Changes with data
Accuracy	Exact under assumptions	Approaches theoretical value as trials increase

In many IB questions, you are asked to compare the two, and explain why they differ. Random variation, bias in trials, and insufficient sample size are common reasons.

在许多IB题目中，你被要求比较两者，并解释它们为何不同。随机波动、试验偏差以及样本量不足是常见原因。

3. 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 of an event tends to stabilise around the theoretical probability. This is the theoretical foundation for using experimental probability as an estimate.

大数定律指出，随着试验次数增加，事件的实验概率会趋向于稳定在理论概率附近。这是使用实验概率进行估计的理论基础。

For instance, if you roll a fair die many times, the experimental probability of rolling a six will get closer and closer to 1/6 as the number of rolls grows.

例如，若反复掷一颗均匀骰子，随着掷数增加，掷出六点的实验概率会越来越接近1/6。

Note that this does not mean short-term “compensation” — if you have had fewer sixes than expected, future sixes are not more likely. Each trial is independent.

请注意，这并不意味着短期“补偿”——如果你掷出的六点少于预期，未来的六点并不会因此更可能。每次试验都是独立的。

4. Relative Frequency Formula | 相对频率公式

Relative frequency is synonymous with experimental probability in most contexts. It is calculated as:

在大多数情境中，相对频率与实验概率同义。其计算公式为：

Relative frequency = f / n

相对频率 = f / n

where f is the frequency of the event and n is the total number of observations. For example, in a survey of 200 students, if 120 prefer online learning, the relative frequency of that preference is 120/200 = 0.6 = 60%.

其中 f 是事件发生的频数，n 是总观测次数。例如，在一项200名学生的调查中，如果有120人偏好在线学习，则该偏好的相对频率为120/200 = 0.6 = 60%。

5. Experiments vs Simulations | 实验与模拟

Experimental probability can be obtained through physical experiments such as tossing coins, rolling dice, or drawing cards. However, many situations are difficult or impossible to repeat physically, so simulations are used instead.

实验概率可以通过物理实验获得，例如抛硬币、掷骰子或抽牌。然而，许多情境难以或无法实际重复，因此可以使用模拟代替。

Simulations use random number generators, spreadsheets, or graphing calculators to model a real-life process. For example, to estimate the probability that at least two people in a class share a birthday, you might simulate 1000 class groups.

模拟利用随机数生成器、电子表格或图形计算器来建模真实过程。例如，要估计一个班级中至少两人同生日概率，可以模拟1000个班级群体。

IB Mathematics encourages the use of technology for such simulations, and you should be able to interpret the results and recognise limitations.

IB数学鼓励使用技术进行此类模拟，你应该能够解读结果并识别局限性。

6. Using Random Number Tables | 使用随机数表

Before graphing calculators and computers were common, random number tables provided a simple way to simulate probabilities. You can assign digits or pairs of digits to outcomes according to their probabilities.

在图形计算器和计算机普及之前，随机数表提供了简单的概率模拟方法。你可以将单个数字或数字对按概率分配给各结果。

For example, to simulate a biased coin with probability 0.7 for heads, you could use the digits 0–6 for heads and 7–9 for tails. Each digit from a table has an equal chance, so this reflects the required 70% heads probability.

例如，要模拟正面概率为0.7的偏硬币，可用数字0–6表示正面，7–9表示反面。表中的每个数字机会均等，因此可体现70%正面的概率要求。

7. Representing Experimental Data | 实验数据的表示

Collected experimental data are often presented in frequency tables, relative frequency tables, bar charts, or pie charts. In IB exams, you may need to construct or interpret these displays.

收集到的实验数据通常以频数表、相对频率表、条形图或饼图呈现。在IB考试中，你可能需要制作或解读这些图形。

Frequency table: counts how often each outcome occurs.
Relative frequency table: shows the proportion of each outcome.
Bar chart: compares frequencies visually.

频数表：统计每个结果出现的次数。
相对频率表：显示每个结果所占的比例。
条形图：直观比较频数。

When reporting results, always include the number of trials because experimental probability without sample size is not meaningful.

汇报结果时，一定要包含试验次数，因为没有样本量的实验概率没有意义。

8. Worked Example: Rolling Two Dice | 示例：掷两颗骰子

Suppose you roll two fair dice 60 times and record the sum. The theoretical probability of a sum of 7 is 6/36 = 1/6 ≈ 0.167. After 60 rolls, you observe a sum of 7 on 9 occasions.

假设你掷两颗均匀骰子共60次并记录其和。和为7的理论概率为6/36 = 1/6 ≈ 0.167。在60次掷后，你观察到和为7出现了9次。

Experimental probability = 9/60 = 0.15

实验概率 = 9/60 = 0.15

The experimental value 0.15 is close to the theoretical 0.167. If you continued rolling for 600 trials, the experimental probability would likely move closer to 0.167.

实验值0.15接近理论值0.167。如果你继续掷600次，实验概率很可能会更接近0.167。

9. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Students often mistake experimental probability for theoretical probability, or they think that more trials guarantee the exact theoretical value. In fact, experimental probability approaches the theoretical value but may never equal it exactly.

学生常把实验概率误当作理论概率，或认为试验次数越多就一定会得到精确的理论值。实际上，实验概率接近理论值，但可能永远不会完全相等。

Do not round too early: keep fractions or decimals with enough precision.
Always state the number of trials when giving an experimental probability.
Check that your simulation is truly random and unbiased.

不要过早四舍五入：保留足够精度的分数或小数。
给出实验概率时，一定要说明试验次数。
检查你的模拟是否真正随机且无偏。

10. Applications in Real Life | 在现实生活中的应用

Experimental probability is widely used in weather forecasting, insurance risk assessment, quality control, and sports statistics. For instance, a baseball player’s batting average is an experimental probability based on past at-bats.

实验概率广泛应用于天气预报、保险风险评估、质量控制和体育统计。例如，棒球运动员的安打率就是一种基于过去打数的实验概率。

In medicine, the effectiveness of a drug is estimated through clinical trials. The observed proportion of patients who recover is an experimental probability that guides treatment decisions.

在医学中，药物的有效性通过临床试验来估计。观察到的患者康复比例就是一种实验概率，用于指导治疗决策。

11. Connection to IB Assessment | 与IB考试的关联

IB exam questions may ask you to carry out a small experiment, compute relative frequencies, compare them with theoretical probabilities, or discuss the effect of sample size. Longer questions may combine experimental probability with expectation, standard deviation, or normal approximation.

IB考试题可能要求你进行小型实验、计算相对频率、与理论概率比较，或讨论样本量的影响。较长的题目可能将实验概率与期望、标准差或正态近似结合。

You should also be able to evaluate different experimental designs and suggest improvements, such as increasing the number of trials or using a more accurate simulation method.

你还应该能够评价不同实验设计并提出改进建议，例如增加试验次数或使用更精确的模拟方法。

12. Summary: From Data to Insight | 总结：从数据到洞见

Experimental probability converts raw observations into meaningful estimates of uncertainty. It helps us make decisions when we do not know the true underlying model, and it reinforces the crucial idea that repeated trials lead to more reliable conclusions.

实验概率将原始观测转化为对不确定性的有意义估计。当我们不知道真实底层模型时，它帮助我们做出决策，并强化了一个关键观念：重复试验带来更可靠的结论。

Always remember: probability is not just about predicting the future — it is about quantifying how likely different futures are, based on evidence.

永远记住：概率不仅仅是预测未来，而是基于证据量化不同未来发生的可能性。

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Experimental Probability in IB Mathematics | IB数学：实验概率的理解与应用