Understanding Probability | 理解概率

📚 Understanding Probability | 理解概率

Probability is a fundamental concept in mathematics that helps us quantify uncertainty. Whether you are predicting the weather, playing a game of chance, or making decisions based on risk, probability provides a way to describe how likely something is to happen. In this article, we will explore the basic ideas of probability, learn how to calculate probabilities for simple and combined events, and look at useful tools like sample spaces and tree diagrams.

概率是数学中的一个基本概念，帮助我们量化不确定性。无论你是在预测天气、玩机会游戏还是基于风险做决策，概率都提供了一种描述某件事情发生可能性的方法。在本文中，我们将探讨概率的基本思想，学习如何计算简单事件和组合事件的概率，并了解样本空间和树形图等有用工具。

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1. What is Probability? | 什么是概率？

Probability is a number that describes how likely an event is to occur. It can be written as a fraction, decimal, or percentage.

概率是一个描述事件发生可能性的数字，可以用分数、小数或百分比表示。

An event with a probability of 0 is impossible, while an event with a probability of 1 is certain to happen.

概率为 0 的事件是不可能发生的，而概率为 1 的事件是必然发生的。

Most events have probabilities between 0 and 1, such as 0.5 (or ½) for an even chance.

大多数事件的概率在 0 和 1 之间，例如 0.5（或 ½）表示等可能的机会。

Probability can be expressed in words: impossible, unlikely, even chance, likely, certain.

概率可以用词语表达：不可能、不太可能、等可能、很可能、必然。

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2. The Probability Scale | 概率尺度

The probability scale places events on a line from 0 (impossible) to 1 (certain). This helps us compare how likely different events are.

概率尺度将事件放置在从 0（不可能）到 1（必然）的直线上，这有助于我们比较不同事件的可能性大小。

• Impossible: probability = 0

  不可能：概率 = 0

• Unlikely: probability close to 0, e.g., 0.1

  不太可能：概率接近 0，例如 0.1

• Even chance: probability = 0.5 (½)

  等可能：概率 = 0.5（½）

• Likely: probability close to 1, e.g., 0.9

  很可能：概率接近 1，例如 0.9

• Certain: probability = 1

  必然：概率 = 1

We often mark events on a probability line to visualise their likelihood.

我们经常在概率线上标记事件，以直观显示它们的可能性大小。

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3. Basic Probability Formula | 基本概率公式

For situations where all outcomes are equally likely, we can use the formula:

对于所有结果等可能的情况，我们可以使用以下公式：

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

Here, P(Event) means the probability of the event happening.

这里，P(Event) 表示事件发生的概率。

For example, when rolling a fair six-sided die, the probability of rolling a 3 is ⅙ because there is 1 favourable outcome and 6 possible outcomes.

例如，掷一个均匀的六面骰子时，掷出 3 的概率是 ⅙，因为有 1 个有利结果和 6 个可能结果。

The probability of rolling an even number (2, 4, or 6) is 3/6 = ½.

掷出偶数（2、4 或 6）的概率是 3/6 = ½。

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4. Experimental Probability | 实验概率

Experimental probability is based on actually carrying out an experiment or collecting data. It is calculated as:

实验概率基于实际进行实验或收集数据，它的计算方法是：

Experimental Probability = Number of times event occurs ÷ Total number of trials

For instance, if you flip a coin 100 times and get heads 48 times, the experimental probability of heads is 48/100 = 0.48.

例如，如果你抛硬币 100 次，得到 48 次正面，那么正面的实验概率是 48/100 = 0.48。

This value might differ from the theoretical probability of ½, especially for a small number of trials. As the number of trials increases, experimental probability usually gets closer to the theoretical probability.

这个值可能与理论概率 ½ 不同，尤其在试验次数较少时。随着试验次数增加，实验概率通常会越来越接近理论概率。

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5. Theoretical Probability | 理论概率

Theoretical probability is what we expect to happen based on equally likely outcomes, without actually doing the experiment. It relies on the formula P(Event) = favourable / total.

理论概率是我们基于等可能结果预期会发生的情况，而不实际进行实验。它依赖于公式 P(事件) = 有利结果数 / 总结果数。

For a fair coin, the theoretical probability of heads is ½. For a fair die, the probability of rolling a number greater than 4 is 2/6 = ⅓.

对于一枚均匀的硬币，正面的理论概率是 ½。对于一个均匀的骰子，掷出大于 4 的数的概率是 2/6 = ⅓。

Theoretical probability is idealised; real-world experiments may show slight variations due to randomness.

理论概率是理想化的；真实世界的实验可能因随机性而显示轻微差异。

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6. Mutually Exclusive Events | 互斥事件

Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a die, rolling a 3 and rolling a 5 are mutually exclusive – a single roll cannot show both numbers.

如果两个事件不能同时发生，则它们互斥。例如，掷骰子时，掷出 3 和掷出 5 是互斥的——单次掷骰不可能同时显示这两个数字。

For mutually exclusive events A and B, the probability that either A or B occurs is given by:

对于互斥事件 A 和 B，事件 A 或 B 发生的概率为：

P(A or B) = P(A) + P(B)

Thus, the probability of rolling a 3 or a 5 is ⅙ + ⅙ = 2/6 = ⅓.

因此，掷出 3 或 5 的概率是 ⅙ + ⅙ = 2/6 = ⅓。

If events are not mutually exclusive, we must subtract the overlap to avoid double-counting, but at KS3 the focus is on mutually exclusive cases.

如果事件不互斥，我们必须减去重叠部分以避免重复计数，但KS3阶段主要关注互斥的情况。

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7. Complementary Events | 互补事件

The complement of an event A is the event that A does not happen, written as A’ or “not A”. Since an event must either happen or not happen, the probabilities of A and its complement add up to 1.

事件 A 的补集是 A 不发生的事件，记作 A’ 或“非 A”。因为一个事件要么发生要么不发生，所以 A 与其补集的概率之和为 1。

P(not A) = 1 − P(A)

For example, if the probability of rain tomorrow is 0.3, then the probability of no rain is 1 − 0.3 = 0.7.

例如，如果明天下雨的概率是 0.3，那么不下雨的概率就是 1 − 0.3 = 0.7。

Using complements can sometimes make probability calculations much simpler.

有时使用补集可以让概率计算变得非常简单。

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8. Sample Spaces and Listing Outcomes | 样本空间与列出结果

A sample space is the set of all possible outcomes of an experiment. Systematic listing helps ensure no outcomes are missed.

样本空间是一个实验所有可能结果的集合。系统性地列出结果有助于确保没有遗漏。

For a single coin toss, the sample space is {Heads, Tails}. For rolling a die, it is {1, 2, 3, 4, 5, 6}.

对于单次抛硬币，样本空间是 {正面，反面}。对于掷骰子，样本空间是 {1, 2, 3, 4, 5, 6}。

When two coins are tossed, the sample space can be listed as: HH, HT, TH, TT. This helps us see that the probability of getting at least one tail is 3/4 = ¾.

当抛两枚硬币时，样本空间可以列为：HH, HT, TH, TT。这有助于我们看到至少得到一次反面的概率是 3/4 = ¾。

For rolling two dice, a table is very useful to list all 36 outcomes:

对于掷两个骰子，使用表格列出所有 36 个结果非常有用：

+	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

The table shows the sum of the two dice. We can see that there are 6 ways to get a sum of 7, so P(sum = 7) = 6/36 = ⅙.

该表显示了两个骰子的和。我们可以看到有 6 种方式得到和为 7，因此 P(和为 7) = 6/36 = ⅙。

Systematic listing and tables are important skills for solving probability problems without missing outcomes.

系统性列举和制作表格是解决概率问题且不遗漏结果的重要技能。

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9. Probability of Combined Events (AND and OR) | 组合事件的概率（和/或）

When we want two independent events to both happen, we multiply their probabilities. Two events are independent if the outcome of one does not affect the outcome of the other.

当我们希望两个独立事件都发生时，我们将它们的概率相乘。如果两个事件中一个的结果不影响另一个的结果，则这两个事件独立。

P(A and B) = P(A) × P(B)

For example, the probability of flipping a head and rolling a 5 on a die is ½ × ⅙ = 1/12.

例如，抛硬币得到正面并且骰子掷出 5 的概率是 ½ × ⅙ = 1/12。

For the probability that event A or event B occurs, we use the addition rule, remembering that if the events are mutually exclusive, simply add the probabilities:

对于事件 A 或事件 B 发生的概率，我们使用加法规则。记住，如果事件互斥，只需将概率相加：

P(A or B) = P(A) + P(B)

Always check whether events are mutually exclusive before using this addition rule.

在使用这个加法规则之前，一定要检查事件是否互斥。

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10. Using Tree Diagrams | 使用树形图

Tree diagrams are extremely helpful for visualising sequences of events, especially when outcomes have different probabilities or when events are repeated.

树形图对于将一系列事件可视化非常有用，尤其是当结果具有不同概率或事件重复发生时。

Each branch of the tree represents a possible outcome, and we write the probability along the branch. The final outcomes are found by multiplying probabilities along the branches.

树中的每条分支代表一个可能的结果，我们沿分支写出概率。最终结果的概率通过将沿途各分支的概率相乘得出。

Consider drawing two marbles from a bag containing 2 red and 3 blue marbles, without replacement. The probabilities on the second draw change depending on the first outcome. A tree diagram makes this clear.

考虑从一个装有 2 个红球和 3 个蓝球的袋子中不放回地抽取两个球。第二次抽取的概率会因第一次结果而改变。树形图能让这一点变得清楚。

For independent events, like flipping a coin twice, the tree is symmetrical:

对于独立事件，例如抛两次硬币，树形图是对称的：

• First flip: Heads (½) or Tails (½)

  第一次抛：正面 (½) 或反面 (½)

• Second flip: Heads (½) or Tails (½) from each first outcome

  第二次抛：从每个第一次结果出发，正面 (½) 或反面 (½)

The probability of getting two heads is ½ × ½ = ¼. The probability of exactly one head is (½ × ½) + (½ × ½) = ½.

得到两个正面的概率是 ½ × ½ = ¼。恰好得到一个正面的概率是 (½ × ½) + (½ × ½) = ½。

Tree diagrams help manage more complicated conditional probabilities as you advance in mathematics.

随着数学学习的深入，树形图有助于处理更复杂的条件概率。

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11. Relative Frequency and Expected Outcomes | 相对频率与预期结果

Relative frequency is another name for experimental probability – it is the ratio of the number of times an event occurs to the total number of trials.

相对频率是实验概率的另一个名称——它是事件发生的次数与试验总次数的比值。

We can use probability to predict how many times an event should occur in a certain number of trials. This is called the expected frequency:

我们可以用概率来预测某个事件在一定的试验次数中应发生的次数，这被称为预期频数：

Expected frequency = Probability × Number of trials

If a die is rolled 300 times, we expect to roll a 6 about ⅙ × 300 = 50 times.

如果掷一个骰子 300 次，我们预期大约有 ⅙ × 300 = 50 次掷出 6。

It is important to remember that expected values are not guarantees – the actual result may differ due to randomness, but over many trials the average should be close.

记住预期值并非保证这一点很重要——实际结果可能因随机性而异，但在大量试验后，平均值应该接近预期值。

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12. Common Misconceptions | 常见误解

One common mistake is believing that previous outcomes affect future independent events – this is called the gambler’s fallacy. If a fair coin lands on heads five times in a row, the probability of heads on the next toss is still ½.

一个常见错误是认为先前的结果会影响未来的独立事件——这被称为赌徒谬误。如果一枚均匀硬币连续五次出现正面，下一次抛掷出现正面的概率仍然是 ½。

Another misconception is adding probabilities without checking if events are mutually exclusive.

另一个误解是在没有检查事件是否互斥的情况下就直接将概率相加。

Students also sometimes confuse P(A and B) with P(A or B). Remember: AND means multiply (for independent events), OR means add (for mutually exclusive events).

学生有时还会混淆 P(A and B) 与 P(A or B)。记住：AND 意味着相乘（针对独立事件），OR 意味着相加（针对互斥事件）。

Finally, always ensure the probabilities of all possible outcomes add up to 1 – this is a useful check.

最后，务必确保所有可能结果的概率之和为 1——这是一个有用的检验方法。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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