Experimental and Theoretical Probability | 实验概率与理论概率

📚 Experimental and Theoretical Probability | 实验概率与理论概率

Understanding probability is essential in mathematics and everyday life. It helps us predict outcomes, make decisions, and evaluate risks. In this article, we will explore the key ideas of probability scales, experimental and theoretical probability, sample spaces, combined events, tree diagrams, expectation, and fair games. Whether you are tossing a coin or planning a strategy, probability provides the tools to measure chance.

理解概率在数学和日常生活中至关重要。它帮助我们预测结果、做出决策和评估风险。在本文中，我们将探讨概率标度、实验概率与理论概率、样本空间、组合事件、树状图、期望值和公平游戏等关键概念。无论你是在抛硬币还是制定策略，概率都提供了衡量可能性的工具。

---

1. Probability Scale | 概率标度

Probability is a measure of how likely an event is to happen. It is always expressed as a number between 0 and 1, inclusive. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain to happen. A probability of 0.5 (or ½) indicates that the event has an even chance of occurring or not occurring.

概率衡量事件发生的可能性，总是用 0 到 1 (含) 之间的数字表示。0 表示事件不可能发生，1 表示事件一定会发生。概率为 0.5 (即 ½) 表示事件发生与不发生的可能性相等。

For example, when you toss a fair coin, the probability of getting Heads is ½, and the probability of getting Tails is also ½. Both outcomes are equally likely. On a probability scale drawn as a number line from 0 to 1, impossible events sit at 0, certain events at 1, and equally likely events at the midpoint. Words such as ‘unlikely’, ‘likely’, ‘even chance’ help describe positions on this scale.

例如，抛一枚公平硬币时，得到正面的概率是 ½，得到反面的概率也是 ½。这两种结果是等可能的。在从 0 到 1 的数轴概率标度上，不可能事件位于 0，必然事件位于 1，等可能事件位于中点。 “不可能”、“可能”、“等可能”等词语有助于描述标度上的位置。

We often write probabilities as fractions, decimals, or percentages. The probability of the Sun rising tomorrow is effectively 1, while the probability of throwing a 7 on a standard six-sided die is 0. Being able to place events on the probability line strengthens our understanding of chance.

我们通常用分数、小数或百分数表示概率。太阳明天升起的概率实际上是 1，而掷一个标准六面骰掷出 7 的概率是 0。能够将事件放在概率线上可以加深我们对可能性的理解。

---

2. Experimental Probability | 实验概率

Experimental probability (or relative frequency) is found by carrying out an experiment or observing actual trials. It is calculated by dividing the number of times the event occurs by the total number of trials. The formula is:

实验概率 (或称相对频率) 通过进行实验或观察实际试验得到。计算方法为事件发生的次数除以总试验次数。公式为：

Experimental probability = Number of times the event occurs ÷ Total number of trials

实验概率 = 事件发生的次数 ÷ 总试验次数

Imagine tossing a coin 100 times and recording 53 Heads. The experimental probability of getting a Head is 53/100 = 0.53. This value may not be exactly 0.5, because the experiment is subject to random variation. In another set of 100 tosses, you might count 47 Heads, giving an experimental probability of 0.47.

假设抛一枚硬币 100 次，记录到 53 次正面。那么得到正面的实验概率为 53/100 = 0.53。该值不一定正好是 0.5，因为实验会受随机变化影响。在另外一组 100 次抛掷中，可能记录到 47 次正面，实验概率就是 0.47。

Experimental probability is useful when theoretical probability cannot be easily determined, such as the probability of a particular footballer scoring a penalty. Coaches and analysts use past data (trials) to estimate the player’s success rate. The more trials collected, the more reliable the experimental probability becomes.

当理论概率不容易确定时，实验概率就很有用，比如某位足球运动员罚中点球的概率。教练和分析师利用过去的数据 (试验) 来估计球员的成功率。收集的试验次数越多，实验概率就越可靠。

---

3. Theoretical Probability | 理论概率

Theoretical probability is determined by reasoning about the possible outcomes, without needing to run an experiment. It assumes that all outcomes in the sample space are equally likely. The formula is:

理论概率通过对所有可能结果的推理得出，无需进行实验。它假定样本空间中所有结果等可能。公式为：

Theoretical probability = Number of favourable outcomes ÷ Total number of possible outcomes

理论概率 = 有利结果的数量 ÷ 所有可能结果的总数

For a fair six-sided die, the possible outcomes are {1, 2, 3, 4, 5, 6}. The event ‘rolling an even number’ has three favourable outcomes {2, 4, 6}. Therefore the theoretical probability is 3/6 = 1/2. Similarly, for a standard deck of 52 playing cards, the probability of drawing an Ace is 4/52 = 1/13, because there are four Aces among 52 equally likely cards.

对一个公平的六面骰，可能的结果是 {1, 2, 3, 4, 5, 6}。事件 ‘掷出偶数’ 有三个有利结果 {2, 4, 6}，所以理论概率是 3/6 = 1/2。同样，对于一副标准的 52 张扑克牌，抽到 A 的概率是 4/52 = 1/13，因为在 52 张等可能的牌中有四张 A。

Theoretical probability always gives a perfect mathematical model, provided the objects involved are fair and unbiased. It is the foundation of many probability problems and allows us to calculate expectations and analyse games.

如果所涉及的物体是公平无偏的，理论概率总能给出一个完美的数学模型。它是许多概率问题的基础，使我们能够计算期望值并分析各种游戏。

---

4. Comparing Experimental and Theoretical Probability | 实验概率与理论概率的比较

When an experiment is performed a small number of times, the experimental probability can differ significantly from the theoretical value. However, as the number of trials increases, the experimental probability tends to get closer to the theoretical probability. This behaviour is explained by the Law of Large Numbers.

当实验只进行少量次数时，实验概率可能与理论值有较大差异。然而，随着试验次数的增加，实验概率往往会趋近于理论概率。大数定律解释了这个现象。

Consider tossing a fair coin. The theoretical probability of Heads is 0.5. If you toss the coin only 10 times, you might see 7 Heads (experimental probability 0.7). But if you toss it 1000 times, the relative frequency is likely to be much closer to 0.5, say 0.498 or 0.503. Computer simulations can demonstrate this convergence beautifully.

以抛公平硬币为例。正面的理论概率是 0.5。如果只抛 10 次，你可能看到 7 次正面 (实验概率 0.7)。但如果抛 1000 次，相对频率很可能会非常接近 0.5，例如 0.498 或 0.503。计算机模拟可以很好地展示这种收敛趋势。

Knowing both types of probability helps students understand that probability is not about guaranteeing a specific short-term outcome, but about describing long-term patterns. In real-life situations such as weather forecasting, probabilities are often based on models (theoretical) refined by historical observations (experimental).

了解两种概率有助于学生明白，概率并不是保证某个具体的短期结果，而是描述长期模式。在天气预报等现实情况中，概率通常基于通过历史观测 (实验) 修正的模型 (理论)。

---

5. Sample Spaces and Equally Likely Outcomes | 样本空间与等可能结果

A sample space is the complete list of all possible outcomes of an experiment or random process. If all outcomes are equally likely, calculating probability becomes straightforward. For a single coin toss, the sample space is {Head, Tail}. For one fair die, it is {1, 2, 3, 4, 5, 6}.

样本空间是实验或随机过程所有可能结果的完整列表。如果所有结果等可能，计算概率就变得简单直接。对抛一枚硬币，样本空间是 {正面, 反面}。对一枚公平骰子，样本空间是 {1, 2, 3, 4, 5, 6}。

When two or more events happen together, we must list all combined outcomes. For flipping two coins, the sample space is {HH, HT, TH, TT}, where H = Head, T = Tail. Notice that HT and TH are considered different outcomes because the order of coins matters, making the total number of outcomes 4. The probability of getting exactly one Head is 2/4 = 1/2.

当两个或多个事件同时发生时，我们必须列出所有组合结果。对于抛两枚硬币，样本空间是 {HH, HT, TH, TT}，其中 H 代表正面，T 代表反面。注意 HT 和 TH 被视为不同的结果，因为硬币的顺序很重要，这样使总结果数为 4。恰好得到一个正面的概率是 2/4 = 1/2。

A systematic list, a table, or a tree diagram ensures that no outcome is missed. Organising outcomes logically is a key skill in KS3 mathematics and prevents common errors when counting favourable outcomes.

有系统的列表、表格或树状图可以确保不会遗漏任何结果。有逻辑地组织结果是 KS3 数学的关键技能，能防止在计算有利结果时出现常见错误。

---

6. Probability of Combined Events – Using Tables | 组合事件的概率 – 使用表格

For two independent events, a two-way table (or possibility space diagram) is extremely useful. Let’s consider rolling two fair six-sided dice and recording the sum of the numbers. The sample space consists of 36 equally likely ordered pairs. The table below shows the sum for each pair.

对于两个独立事件，双向表格 (或可能性空间图) 非常有用。考虑同时掷两个公平六面骰并记录点数之和。样本空间由 36 个等可能的有序数对组成。下表显示了每个数对的和。

+	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

To find the probability that the sum is exactly 7, count how many times 7 appears in the table: six times. So P(sum = 7) = 6/36, which simplifies to 1/6. Similarly, you can quickly read off P(sum ≥ 10) by counting the cells with 10, 11 or 12: there are 6 favourable outcomes, so P = 6/36 = 1/6.

要找出和为 7 的概率，数一数表格中 7 出现了多少次：六次。因此 P(和为7) = 6/36 = 1/6。同样，你可以快速读出 P(和 ≥ 10) 只需数 10、11、12 的格子数：共有 6 个有利结果，所以 P = 6/36 = 1/6。

This tabular method is powerful for two-event problems and helps avoid double-counting or missing outcomes. It directly connects to the concept of sample spaces and prepares students for more advanced probability tools like Venn diagrams.

这种表格方法对双事件问题非常有效，有助于避免重复计数或遗漏结果。它直接与样本空间概念联系起来，并为进一步学习韦恩图等更高级的概率工具做好准备。

---

7. Tree Diagrams for Independent Events | 独立事件的树状图

A tree diagram visually organises the possible outcomes of two or more independent events. Each branch is labelled with the probability of that outcome. To calculate the probability of a sequence of events, multiply the probabilities along the branches (the AND rule).

树状图直观地组织两个或多个独立事件的可能结果。每条分支都标有该结果的概率。要计算一系列事件发生的概率，只需沿分支相乘 (AND 法则)。

For example, a bag contains 3 red (R) and 5 blue (B) marbles. You pick one, then replace it and pick another. The first branch: P(R) = 3/8 and P(B) = 5/8. Because of replacement, the second pick has the same probabilities. The probability of picking Red first and Red second = 3/8 × 3/8 = 9/64. The probability of Red then Blue = 3/8 × 5/8 = 15/

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

更多咨询请联系16621398022（同微信）