📚 A - Experimental Probability: Definition, Formula and Worked Examples | A - 实验概率：定义、公式与例题解析
Experimental probability is a foundational concept in IB Mathematics that connects theoretical models with real-world data. Through hands-on experiments, simulations and data analysis, students explore how probability estimates become more accurate as the number of trials increases. This article covers the key definitions, formulas, worked examples and applications of experimental probability, fully aligned with IB learning objectives and equally useful for A-Level and GCSE students.
实验概率是IB数学中的一个基本概念，它架起了理论模型与真实世界数据之间的桥梁。通过动手实验、模拟和数据分析，学生探索随着试验次数的增加，概率估计如何变得更加精确。本文涵盖实验概率的关键定义、公式、例题解析与应用，既与IB学习目标保持一致，也适用于A-Level与GCSE阶段的学生。
1. What is Experimental Probability? | 什么是实验概率？
Experimental probability, also called empirical probability, is the probability of an event obtained by actually carrying out an experiment or a set of trials, observing the outcomes, and counting how often the event of interest occurs. You perform the experiment many times, record the results, and divide the number of times the event happened by the total number of trials. Each single repetition of the experiment is called a trial, and each possible result of a trial is called an outcome.
实验概率（又称经验概率）是通过实际进行实验或一系列试验、观察结果，并统计目标事件出现的次数而得到的概率。你需要多次重复实验、记录结果，再用事件发生的次数除以总试验次数。实验的每一次重复称为一次试验（trial），而一次试验的每一种可能结果称为一个结果（outcome）。
For example, if you flip a fair coin 50 times and get heads 23 times, the experimental probability of heads is 23/50 = 0.46, which is 46%. No theory is used at all - the estimate comes purely from the data you collected.
例如，如果你抛一枚硬币50次，得到23次正面，那么正面朝上的实验概率为 23/50 = 0.46，即46%。这里完全没有用到理论推导——估计值完全来自你收集到的数据。
2. Experimental vs Theoretical Probability | 实验概率与理论概率
outcomes by the total number of equally likely outcomes, so for a fair coin the theoretical probability of heads is 1/2 = 0.5.
理论概率基于"当所有结果等可能时应该发生什么"。它的计算方式是：有利结果数除以所有等可能结果的总数，因此对于一枚公平硬币，正面朝上的理论概率为 1/2 = 0.5。
Unlike theoretical probability, which is based on what should happen, experimental probability is based on what actually did happen. It provides a real-world estimate that can be compared with the theoretically expected value. Because of random variation, the two values are rarely exactly equal after a small number of trials, but they usually get closer as the experiment is repeated more often.
与理论概率（基于应该发生的情况）不同，实验概率基于实际已经发生的情况。它提供了一个现实世界的估计值，可以与理论期望值进行比较。由于随机波动的存在，在试验次数较少时两者很少完全相等，但随着实验不断重复，它们通常会越来越接近。
| Aspect 对比项 | Theoretical 理论概率 | Experimental 实验概率 |
| Source 依据 | Reasoning, symmetry, equally likely outcomes 推理、对称性、等可能结果 | Observed data from trials 试验中观察到的数据 |
| Formula 公式 | favourable ÷ equally likely outcomes | frequency of event ÷ number of trials |
| Coin example 硬币示例 | 1/2 = 0.5 | 23/50 = 0.46 |
| Stability 稳定性 | Fixed, exact 固定、精确 | Varies, improves with more trials 会波动，试验越多越稳定 |
3. The Core Formula | 核心公式
The formula for experimental probability is short and universal. Write f for the number of times the event occurred (its frequency) and n for the total number of trials, and you have:
实验概率的公式简短而且通用。用 f 表示事件发生的次数（频数），用 n 表示试验的总次数，则有：
Pexp(A) = f / n = (number of times A occurred) ÷ (total number of trials)
low from this definition. First, the value always lies between 0 and 1 inclusive, so 0 ≤ Pexp ability of 0 means the event was not observed in that particular experiment, not that it is impossible. Third, if the outcomes of an experiment are mutually exclusive and exhaustive, then the experimental probabilities of all outcomes add up to 1.
由这个定义可以立刻得到三条规则。第一，取值始终在0到1之间（含端点），即 0 ≤ Pexp(A) ≤ 1。第二，实验概率为0只表示在该次实验中未观察到该事件，并不代表它不可能发生。第三，如果一次实验的各结果互斥且完备，那么所有结果的实验概率之和等于1。
xpected to convert between fractions, decimals and percentages fluently: 23/50 = 0.46 = 46%. Always give the final answer in the form requested by the question.
你还需要在分数、小数与百分数之间熟练转换：23/50 = 0.46 = 46%。请务必按照题目要求的形式给出最终答案。
4. Worked Example: The Coin Toss | 例题详解：抛硬币
ility of tails is 27/50 = 0.54. Notice that 0.46 + 0.54 = 1, which confirms the calculation, because heads and tails are the only possible outcomes.
某学生抛硬币50次，记录到23次正面和27次反面。正面朝上的实验概率为 23/50 = 0.46，反面朝上的实验概率为 27/50 = 0.54。注意 0.46 + 0.54 = 1，这正好验证了计算，因为正面和反面是仅有的两种可能结果。
The theoretical probability for a fair coin is 0.5 for each side. The experimental value differs from it by only 0.04, which is a small discrepancy for 50 trials. If the student repeated the experiment 500 times, the estimate would very likely move even closer to 0.5, although it would almost certainly not be exactly 0.5.
对于公平硬币，每一面的理论概率都是0.5。实验值与之仅相差0.04，对50次试验来说这是一个很小的偏差。如果该学生再做500次试验，估计值很可能会更接近0.5，尽管几乎肯定不会恰好等于0.5。
Exam tip: when a question says "estimate the probability" or "from the data", it expects an experimental answer expressed as a fraction or decimal of the form f/n. When a question describes a fair die, a fair coin or a perfectly symmetrical spinner, it expects a theoretical answer.
考试提示：当题目说"估计概率"或"根据数据"，它要的是形如 f/n 的分数或小数形式的实验答案；当题目描述的是公平骰子、公平硬币或完全对称的转盘时，它要的是理论答案。
5. Relative Frequency and the Law of Large Numbers | 相对频率与大数定律
Experimental probability is sometimes called relative frequency, because it measures how frequently an event occurs relative to the total number of trials. The key theoretical idea behind it is the Law of Large Numbers: as the number of trials n increases, the relative frequency tends to get closer and closer to the theoretical probability.
实验概率有时也被称为相对频率，因为它衡量的是事件发生的次数相对于总试验次数的比例。其背后的核心理论思想是大数定律：当试验次数 n 增大时，相对频率会越来越接近理论概率。
Pexp(A) → P(A) as n → ∞
This is exactly why simulations, spreadsheets and computer programs are so valuable in probability: they let you run thousands or millions of trials in seconds, giving estimates that are far more stable than any 20-toss classroom experiment.
这正是模拟、电子表格和计算机程序在概率学习中如此有价值的原因：它们可以在几秒内运行成千上万次甚至上百万次试验，得到的估计值远比课堂上的20次抛硬币稳定得多。
Beware of the gambler's fallacy. The Law of Large Numbers does not mean that a coin "owes" you tails after a run of heads; each trial remains independent, and the convergence comes from the growing denominator, not from any correction mechanism.
要警惕"赌徒谬误"。大数定律并不意味着硬币在连续出现正面后就"欠"你一次反面；每次试验仍然是独立的，趋同来自不断增大的分母，而不是任何自我修正机制。
6. Designing a Fair Experiment | 如何设计一个公平的实验
A badly designed experiment produces a biased estimate, no matter how long the table of results looks. Before collecting data, be clear about the trial, the event of interest and the conditions that must stay constant.
设计糟糕的实验会产生有偏的估计，无论结果表格看起来多长都无济于事。在收集数据之前，必须明确什么是试验、什么是目标事件，以及哪些条件必须保持不变。
- Define the trial precisely - one toss, one spin, one draw. 精确定义试验——一次抛掷、一次旋转、一次抽取。
- Make every trial genuinely random: shake the dice cup, spin the spinner properly, use replacement when drawing. 确保每次试验真正随机：摇匀骰盅、正确旋转转盘、抽取时放回。
- Decide the number of trials in advance and use as many as practical; 100 or more is usually reasonable. 事先确定试验次数并尽可能多做，通常100次以上较为合理。
- Keep conditions constant: the same coin, die or spinner, and the same method each time. 保持条件不变：使用同一枚硬币、同一颗骰子或同一个转盘，每次方法一致。
- Record results honestly in a tally table before calculating anything. 在计算之前用划记表如实记录结果。
- Repeat the whole experiment if the results look suspiciously extreme. 若结果异常极端，重复整个实验。
An unfair method - such as always starting a dice roll from the same side facing up, or letting students choose whether to participate - introduces bias that more trials will not remove.
不公平的方法——例如每次掷骰子都让同一面朝上开始，或让学生自己决定是否参与——会引入偏差，而这种偏差不会因为增加试验次数而消除。
7. Simulations and Random Number Generators | 模拟与随机数生成器
When a physical experiment is slow, expensive or impossible, a simulation can replace it. A simulation is a model of the real experiment in which outcomes are generated by a random mechanism, such as a spreadsheet function, a graphical calculator command or a short computer program.
当实物实验太慢、成本太高或根本无法进行时，可以用模拟来替代。模拟是真实实验的模型，其结果由某种随机机制生成，例如电子表格函数、图形计算器命令或一段简短的计算机程序。
- : =RANDBETWEEN(1,6) simulates one fair die roll; =RAND() gives a random decimal in [0, 1). 电子表格：=RANDBETWEEN(1,6) 模拟一次公平掷骰；=RAND() 给出 [0, 1) 内的随机小数。
- Calculator: the random functions Ran# and RanInt generate random numbers for quick class demonstrations. 计算器：随机函数 Ran# 与 RanInt 可生成随机数，适合课堂快速演示。
- Programming or statistical software: loops allow tens of thousands of trials and automatic frequency counting. 编程或统计软件：用循环可以完成数万次试验并自动统计频数。
Simulations have limitations. Pseudo-random generators are deterministic algorithms, so they are only "random enough", and a simulation is only as good as the model behind it - if the model assumes a fair die, the simulation will happily confirm a fair die.
模拟也有局限。伪随机生成器本质上是确定性算法，因此只是"足够随机"；模拟的可靠性取决于背后的模型——如果模型假设骰子是公平的，模拟自然会得出公平骰子的结论。
8. More Worked Examples: Dice, Spinners and Quality Control | 更多例题：骰子、转盘与质量控制
/6 ≈ 0.167, so the die looks slightly biased, but 200 trials are not enough to prove it - a formal test would be needed.
例A——一颗骰子掷200次，数字6出现29次。出现6的实验概率为 29/200 = 0.145。理论值为 1/6 ≈ 0.167，因此这颗骰子看起来略偏，但200次试验还不足以证明这一点，需要做正式的检验。
mated probability that a randomly chosen item is defective is 17/500 = 0.034, so about 3.4% of output is faulty.
例B——某工厂从生产线上抽检500件产品，发现17件有缺陷。随机抽取一件产品有缺陷的估计概率为 17/500 = 0.034，即约3.4%的产品不合格。
Example C - a biased coin is tossed 80 times and lands on heads 36 times. Estimate the probability of heads, and predict the number of heads in 400 tosses.
例C——一枚有偏硬币抛80次，其中36次正面朝上。估计正面朝上的概率，并预测抛400次时正面的次数。
show both the probability and the multiplication when the question asks for a prediction.
估计值为 36/80 = 0.45。抛400次时，期望次数为 0.45 × 400 = 180 次正面。当题目要求预测时，务必同时写出概率与乘法运算过程。
9. Estimating Probability from Frequency Tables | 从频率表估计概率
Exam questions often provide grouped or ungrouped frequency data instead of a description of an experiment. The method is unchanged: divide each frequency by the total frequency to obtain the experimental probability of that category.
考试题常给出分组或未分组的频数数据，而不是实验过程的描述。方法不变：用每个频数除以总频数，即可得到该类别的实验概率。
Example: 120 students were asked their favourite subject.
例：询问120名学生最喜欢的科目。
| Subject 科目 | Frequency 频数 | Experimental probability 实验概率 |
| Mathematics 数学 | 42 | 42/120 = 0.35 |
| Science 科学 | 36 | 36/120 = 0.30 |
| English 英语 | 27 | 27/120 = 0.225 |
| Other 其他 | 15 | 15/120 = 0.125 |
| Total 合计 | 120 | 1.000 |
The probabilities sum to 1, which is a useful check on your arithmetic. If the data are given in intervals, such as times or heights, the same idea applies to each interval, and the estimate refers to the class rather than to an exact value.
各概率之和为1，这是检验计算是否正确的好方法。如果数据以区间形式给出（例如时间或身高），同样对每个区间应用该方法，此时估计值针对的是这一组区间，而非某个精确数值。
10. Common Errors and Exam Technique | 常见错误与应试技巧
Most lost marks in this topic come from a small number of predictable mistakes, so check your work against this list before moving on.
本主题失分大多来自少数几个可预见的错误，完成作答前请对照以下清单检查。
- Reversing the fraction: the frequency of the event goes on top, the total number of trials goes on the bottom. 分数写反：事件频数在分子，总试验次数在分母。
- Using the number of outcomes instead of the number of trials in the denominator. 分母误用"结果种类数"而不是"试验次数"。
- Giving a theoretical answer (1/6) when the question asks for an estimate from data. 在要求根据数据估计时，给出理论答案（1/6）。
- Rounding too early; keep full accuracy until the final line. 过早四舍五入；应保留完整精度到最后一行为止。
- Claiming an event is impossible because its experimental probability is 0 after a few trials. 因为在少量试验中实验概率为0，就断言该事件不可能发生。
- Ignoring the requested form - fraction, decimal or percentage - and failing to simplify. 忽略题目要求的分数、小数或百分数形式，且未约分。
- For a prediction question, forgetting to multiply the estimated probability by the new number of trials. 在预测题中忘记用估计概率乘以新的试验次数。
A reliable exam routine is: identify the event, identify the total number of trials, write the fraction, convert, then interpret the answer in the context of the question.
可靠的应试流程是：确定事件、确定总试验次数、写出分数、转换为所需形式，最后结合题目情境解释答案。
11. Real-World Applications and IB Investigation Ideas | 现实应用与IB探究选题
Experimental probability is used wherever real data must be turned into a usable estimate. Weather forecasts quote the relative frequency of rain on similar days, insurers estimate accident rates from claims history, pharmaceutical trials estimate the proportion of patients who respond to a treatment, and factories use sample inspection to control quality.
凡是需要把真实数据转化为可用估计的地方，都会用到实验概率。天气预报引用相似天气条件下降雨的相对频率，保险公司根据理赔历史估计事故率，药物试验估计对治疗有反应的患者比例，工厂则用抽样检验来控制质量。
For an IB internal assessment or an extended investigation, strong topics include testing whether a die or spinner is fair by comparing experimental and theoretical probabilities over several increasing numbers of trials, using a spreadsheet simulation to illustrate the Law of Large Numbers, or estimating the probability of a real-world event such as a bus arriving late.
对于IB内部评估或拓展探究，优秀的选题包括：通过在不同递增试验次数下比较实验概率与理论概率，检验骰子或转盘是否公平；用电子表格模拟演示大数定律；或估计诸如公交车晚点这类真实事件的概率。
A high-scoring investigation states the hypothesis, justifies the sample size, presents raw data in tables, plots the running relative frequency against the number of trials, and discusses sources of bias and uncertainty.
高分探究报告应包含：陈述假设、论证样本量的合理性、用表格呈现原始数据、绘制"累计相对频率—试验次数"图，并讨论偏差来源与不确定性。
12. Summary and Key Takeaways | 小结与要点回顾
Experimental probability turns observations into an estimate. It is complementary to theoretical probability, not a replacement for it: theory tells you what to expect, data tell you what happened, and the Law of Large Numbers links the two.
实验概率把观察转化为估计。它与理论概率是互补关系，而非替代关系：理论告诉你期望什么，数据告诉你实际发生了什么，而大数定律把两者联系起来。
- Experimental probability = frequency of the event ÷ total number of trials. 实验概率 = 事件频数 ÷ 总试验次数。
- It is based on what actually did happen; theoretical probability is based on what should happen. 它基于实际发生的情况，而理论概率基于应该发生的情况。
- The estimate improves as the number of trials increases, but never gains a memory of past trials. 试验次数越多估计越精确，但试验本身不会"记住"过去的结果。
- Fair design, honest recording and enough trials are essential for a trustworthy estimate. 公平的设计、如实的记录与充分的试验次数是可靠估计的必要条件。
- Simulations and random number generators allow very large numbers of trials quickly. 模拟与随机数生成器可以快速完成极大规模的试验。
Pexp(A) = f / n, and Pexp(A) → P(A) as n → ∞
Master these two statements and the worked examples above, and you will handle any experimental probability question that appears in an IB, A-Level or GCSE examination.
掌握以上两条结论以及本文的例题，你就能从容应对IB、A-Level或GCSE考试中出现的任何实验概率题目。
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