Mastering Probability: Experimental and Theoretical Probabilities | 掌握概率：实验概率与理论概率

📚 Mastering Probability: Experimental and Theoretical Probabilities | 掌握概率：实验概率与理论概率

Probability is a branch of mathematics that deals with the likelihood of events occurring. It is widely used in everyday life, from weather forecasting to games of chance, and it helps us make informed decisions under uncertainty. In this article, we will explore the fundamental concepts of probability, distinguish between experimental and theoretical probabilities, and learn how to calculate probabilities for single and combined events.

概率是数学的一个分支，研究事件发生的可能性。它广泛应用于日常生活中，从天气预报到机会游戏，帮助我们在不确定的情况下做出明智的决策。在本文中，我们将探讨概率的基本概念，区分实验概率和理论概率，并学习如何计算单一事件和组合事件的概率。

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1. Introduction to Probability | 概率简介

Probability is defined as a measure of how likely an event is to happen. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means it is certain. Probabilities can also be written as fractions, decimals, or percentages. For example, when you flip a fair coin, the probability of getting heads is 0.5, or 1/2, or 50%.

概率被定义为衡量事件发生可能性的度量。它用介于 0 和 1 之间的数字表示，0 表示不可能发生，1 表示必然发生。概率还可以写成分数、小数或百分比。例如，抛掷一枚公平硬币时，得到正面的概率是 0.5，或 1/2，或 50%。

Probability is based on the concept of random experiments, which are processes that produce outcomes that cannot be predicted with certainty. Each possible result is called an outcome, and the set of all possible outcomes is called the sample space. For rolling a fair six-sided die, the sample space is {1, 2, 3, 4, 5, 6}, and each outcome has an equal chance of occurring if the die is fair.

概率基于随机实验的概念，随机实验是产生无法确切预测结果的过程。每个可能的结果称为一个结果，所有可能结果的集合称为样本空间。对于掷一个公平的六面骰子，样本空间是 {1, 2, 3, 4, 5, 6}，如果骰子是公平的，每个结果发生的可能性相等。

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2. Probability Scale | 概率标度

The probability scale provides a visual way to understand the likelihood of events. Words such as impossible, unlikely, even chance, likely, and certain are placed on a line from 0 to 1. This helps us describe and compare probabilities without using numbers.

概率标度提供了一种直观理解事件可能性的方法。诸如“不可能”、“不太可能”、“一半机会”、“很可能”和“一定”等词语被放置在从 0 到 1 的线上。这有助于我们在不使用数字的情况下描述和比较概率。

Word	Probability Value
Impossible	0
Unlikely	approximately 0.1 to 0.4
Even chance	0.5
Likely	approximately 0.6 to 0.9
Certain	1

For instance, if a weather forecast says there is a 70% chance of rain, this falls into the ‘likely’ category on the probability scale. The probability is 0.7 as a decimal, and it indicates a strong chance of the event happening.

例如，如果天气预报说明天下雨的可能性是 70%，这在概率标度上属于“很可能”类别。该概率用小数表示为 0.7，表明事件发生的可能性很大。

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3. Experimental Probability | 实验概率

Experimental probability, also known as relative frequency, is calculated based on the results of an actual experiment or historical data. You find it by dividing the number of times a specific event occurs by the total number of trials or observations. The formula can be written as:

实验概率，也称为相对频率，是基于实际实验或历史数据的结果计算得出的。它通过将特定事件发生的次数除以试验或观察的总次数求得。公式可以写成：

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

Suppose you toss a coin 100 times and record 48 heads. The experimental probability of getting heads is 48/100, which simplifies to 0.48 or 48%. Notice it is not exactly 0.5 because experiments involve randomness and variation.

假设你抛一枚硬币 100 次，记录到 48 次正面。得到正面的实验概率是 48/100，化简为 0.48 或 48%。请注意它并不完全等于 0.5，因为实验涉及随机性和变化。

As the number of trials increases, experimental probability tends to get closer to the true theoretical probability. This is known as the law of large numbers. In the short term, results can vary widely, but over many trials they stabilise.

随着试验次数的增加，实验概率往往会趋近于真实的理论概率。这就是大数定律。短期内结果可能差异很大，但经过大量试验后它们会稳定下来。

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4. Theoretical Probability | 理论概率

Theoretical probability is determined by reasoning about all equally likely outcomes without performing an experiment. It assumes all outcomes in the sample space have the same chance of occurring. The formula for theoretical probability is:

理论概率是通过推理所有等可能结果来确定的，无需进行实验。它假设样本空间中的所有结果发生的可能性相同。理论概率的公式是：

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

P(A) = 有利结果的数量 ÷ 可能结果的总数

For example, when rolling a fair six-sided die, the event of rolling an even number has three favourable outcomes (2, 4, 6) out of six possible outcomes. Therefore, P(even) = 3/6 = 1/2 = 0.5. Similarly, the probability of rolling a number greater than 4 is 2/6 = 1/3, because only 5 and 6 satisfy the condition.

例如，掷一个公平的六面骰子时，掷出偶数的事件有 3 个有利结果（2, 4, 6），可能结果总共有 6 个。因此，P(偶数) = 3/6 = 1/2 = 0.5。同样，掷出大于 4 的数的概率是 2/6 = 1/3，因为只有 5 和 6 满足条件。

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

Key differences exist between experimental and theoretical probability. Theoretical probability gives what we expect to happen in an ideal, mathematical model, while experimental probability tells us what actually happened in a real test. For a fair coin, theoretical probability of heads is 0.5, but after 20 flips you might get 11 heads, giving an experimental probability of 0.55.

实验概率和理论概率之间存在关键差异。理论概率给出的是在理想化数学模型下我们预期发生的情况，而实验概率则告诉我们实际测试中发生了什么。对于公平硬币，正面的理论概率为 0.5，但抛 20 次后你可能得到 11 次正面，实验概率为 0.55。

It is important to understand that a small number of trials often produces experimental probabilities that differ from theoretical ones. Students sometimes mistakenly expect the experimental result to match theory even for a small sample. By conducting more trials—say 500 coin flips—the experimental probability will typically be much closer to 0.5.

重要的是要理解，少量试验次数通常会产生与理论值不同的实验概率。学生们有时错误地期望即使样本很小，实验结果也能与理论匹配。通过进行更多试验——比如抛 500 次硬币——实验概率通常会更加接近 0.5。

This behaviour is explained by the law of large numbers: as the number of trials increases, the experimental relative frequency approaches the theoretical probability. This bridging concept is fundamental in statistics and helps us trust predictions based on large data sets.

这一行为可以用大数定律来解释：随着试验次数的增加，实验相对频率会趋近于理论概率。这一桥梁概念在统计学中至关重要，并帮助我们信任基于大数据集的预测。

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6. Sample Space and Outcomes | 样本空间与结果

A sample space is a list or diagram that shows all possible outcomes of a random experiment. For a single event like tossing a coin, the sample space is straightforward: {Heads, Tails}. When two coins are tossed together, we can use a systematic list or a tree diagram to organise the four outcomes: {HH, HT, TH, TT}.

样本空间是一个列出或图示展示随机实验所有可能结果的集合。对于像抛一枚硬币这样的单一事件，样本空间很简单：{正面，反面}。当同时抛两枚硬币时，我们可以使用系统列表或树状图来组织四种结果：{HH, HT, TH, TT}。

Tree diagrams are especially useful for multi-stage experiments. Each branch represents a possible outcome at that stage, and by following the branches you can find all combined outcomes. Knowing the sample space allows you to calculate theoretical probability precisely: for two fair coins, P(two heads) = 1/4, since only one outcome out of four is HH.

树状图对于多阶段实验特别有用。每个分支代表该阶段的一个可能结果，通过追踪分支你可以找到所有组合结果。知道样本空间后，你可以精确计算理论概率：对于两枚公平硬币，P(两个正面) = 1/4，因为四个结果中只有一个是 HH。

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7. Probability of Single Events | 单一事件的概率

Calculating the probability of a single event is straightforward when we can count favourable outcomes and total outcomes. For example, a bag contains 3 red marbles and 2 blue marbles. If you pick one marble at random, the probability of picking a red marble is:

当我们能够数出有利结果和总结果时，计算单一事件的概率非常直接。例如，一个袋子里有 3 个红色弹珠和 2 个蓝色弹珠。如果你随机摸出一个弹珠，摸到红色弹珠的概率是：

P(Red) = 3 ÷ 5 = 3/5 = 0.6

P(红色) = 3 ÷ 5 = 3/5 = 0.6

An important property is that the sum of probabilities of all possible mutually exclusive and exhaustive outcomes is 1. In the marble example, P(Red) + P(Blue) = 3/5 + 2/5 = 1. This means the probability that the chosen marble is not red (the complement) is 1 − P(Red) = 2/5.

一个重要的性质是：所有可能且互斥、穷尽的结果的概率之和为 1。在弹珠例子中，P(红色) + P(蓝色) = 3/5 + 2/5 = 1。这意味着选出的弹珠不是红色的概率（互补事件）为 1 − P(红色) = 2/5。

You can also express probabilities as percentages. The above result means you have a 60% chance of drawing a red marble and a 40% chance of drawing a blue one. This makes it easy to compare the likelihood of different outcomes.

你也可以将概率表示为百分比。上述结果意味着你有 60% 的机会摸到红色弹珠，有 40% 的机会摸到蓝色弹珠。这使得比较不同结果的可能性变得容易。

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8. Probability of Combined Events | 组合事件的概率

When two or more events happen together, we are interested in the probability of combined events. For example, rolling two fair dice and adding the scores produces a new set of outcomes. The sample space can be displayed in a two-way table showing all 36 possible pairs. The sum can range from 2 to 12.

当两个或更多事件同时发生时，我们关心组合事件的概率。例如，掷两个公平骰子并将点数相加，产生了一组新的结果。样本空间可以用一个双向表格展示，显示所有 36 种可能的组合。总和可以从 2 到 12。

+	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

From the table we can find, for instance, P(sum = 7) = 6/36 = 1/6, because there are six combinations that give a total of 7. For other combined events, like getting two even numbers, we would count the number of favourable cells and divide by 36.

从表格中我们可以找出，例如 P(总和 = 7) = 6/36 = 1/6，因为有六种组合得到总和为 7。对于其他组合事件，例如得到两个偶数，我们会数出有利单元格的数量并除以 36。

Listing all outcomes systematically eliminates guesswork and ensures that probabilities are accurate. This method is essential for more complex situations and forms the basis of probability trees in later study.

系统地列出所有结果可以消除猜测，并确保概率的准确性。这种方法对于更复杂的情况至关重要，并为后续学习中的概率树奠定了基础。

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9. Mutually Exclusive Events | 互斥事件

Two events are mutually exclusive if they cannot happen at the same time. For example, when a single die is thrown, the events ‘rolling a 3’ and ‘rolling a 5’ are mutually exclusive because you cannot roll a 3 and a 5 simultaneously. However, ‘rolling an even number’ and ‘rolling a number greater than 4’ are not mutually exclusive (the number 6 satisfies both).

两个事件如果不可能同时发生，则称为互斥事件。例如，抛一个骰子时，“掷出 3”和“掷出 5”是互斥的，因为你不能同时掷出 3 和 5。然而，“掷出偶数”和“掷出大于 4 的数”不是互斥事件（数字 6 同时满足两个条件）。

For mutually exclusive events A and B, the probability that either A or B occurs is the sum of their individual probabilities: P(A or B) = P(A) + P(B). For the die, P(3) = 1/6 and P(5) = 1/6, so P(3 or 5) = 1/6 + 1/6 = 2/6 = 1/3.

对于互斥事件 A 和 B，A 或 B 发生的概率是它们各自概率之和：P(A 或 B) = P(A) + P(B)。对于骰子，P(3) = 1/6，P(5) = 1/6，因此 P(3 或 5) = 1/6 + 1/6 = 2/6 = 1/3。

This addition rule is powerful, but it applies only when events cannot occur together. If events can overlap, you must subtract the probability of the overlap to avoid double‑counting, which is covered in more advanced probability topics.

这个加法规则非常有用，但它仅适用于事件不会同时发生的情况。如果事件可以重叠，你必须减去重叠部分的概率以避免重复计算，这一问题将在更高级的概率主题中探讨。

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10. Using Probability to Make Predictions | 运用概率做预测

Probability is not just about calculating past outcomes; it is a tool for making predictions about the future. If we know the theoretical probability of an event, we can estimate how many times it should occur in a given number of trials. For instance, if the probability of winning a game is 0.2, and you play 50 times, you would expect to win about 50 × 0.2 = 10 times.

概率不仅仅是计算过去的结果，更是预测未来的工具。如果我们知道一个事件的理论概率，我们可以估计它在给定的试验次数中应该发生多少次。例如，如果赢得一场游戏的概率是 0.2，你玩 50 次，你会期望赢大约 50 × 0.2 = 10 次。

This expected frequency is not a guarantee; actual results can vary because of randomness. However, the prediction gives a central value that helps us plan and understand what might happen over many attempts. This idea is used in quality control, insurance, and sports analytics.

这个期望频率并非保证，实际结果可能由于随机性而有所不同。然而，这个预测提供了一个中心值，帮助我们规划和理解在多次尝试中可能发生的情况。这一思想应用于质量控制、保险和体育分析中。

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

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