Understanding Probability: The Basics | 理解概率：基础知识

📚 Understanding Probability: The Basics | 理解概率：基础知识

This guide covers the key ideas from page 190 of the Cambridge KS3 Mathematics textbook, focusing on the fundamentals of probability. Probability helps us to measure and describe how likely events are to happen in everyday life, from games of chance to predicting outcomes in science and statistics.

本指南涵盖 Cambridge KS3 数学教材第 190 页的核心内容，重点是概率的基础知识。概率帮助我们衡量和描述日常生活中事件发生的可能性，从机会游戏到科学和统计中的结果预测，都离不开概率。

1. What is Probability? | 什么是概率？

Probability is a branch of mathematics that gives us a way of quantifying uncertainty. It assigns a numerical value between 0 and 1 to an event, where 0 indicates impossibility and 1 indicates certainty.

概率是数学的一个分支，它为我们提供了量化不确定性的方法。它将 0 到 1 之间的一个数值赋予事件，其中 0 表示不可能，1 表示必然。

For example, tossing a fair coin and getting ‘heads’ has a probability of 1/2. The probability that the sun will rise tomorrow is very close to 1, whereas the probability of rolling a 7 on a standard six-sided dice is exactly 0.

例如，抛掷一枚均匀硬币得到“正面”的概率为 1/2。明天太阳升起的概率非常接近 1，而用一颗标准的六面骰子掷出 7 点的概率恰好为 0。

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2. Probability Scale | 概率标度

The probability scale is a visual way of representing how likely events are. It runs from 0 on the left to 1 on the right. We can label positions along the scale with words such as impossible, unlikely, even chance, likely and certain.

概率标度是一种直观表示事件发生可能性的方法。它从左端的 0 延伸到右端的 1。我们可以用不可能、不太可能、等可能性、很可能和必然等词语在标度上进行标注。

An even chance corresponds to a probability of 0.5, meaning the event is just as likely to happen as not. Probabilities less than 0.5 are described as unlikely, while those greater than 0.5 are likely.

等可能性对应的概率是 0.5，意思是事件发生与不发生的可能性刚好相等。概率小于 0.5 的事件被描述为不太可能，而大于 0.5 的则被视为很可能。

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3. Language of Probability | 概率用语

In KS3 mathematics, you are expected to use the correct language to describe probability. Words such as ‘certain’, ‘even chance’ and ‘impossible’ must be linked to their numerical values or ranges.

在 KS3 数学中，你需要使用正确的语言来描述概率。“必然”、“等可能性”和“不可能”等词语必须与它们的数值或取值范围联系起来。

Other common descriptors include ‘very unlikely’ for probabilities near 0, ‘likely’ for those above 0.5, and ‘very likely’ for events with probabilities close to 1. Being precise with language helps you to communicate your understanding clearly.

其他常用的描述词包括“非常不可能”（概率接近 0）、“很可能”（大于 0.5）以及“非常可能”（概率接近 1）。精确使用语言有助于你清晰地表达自己的理解。

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4. Calculating Simple Probabilities | 计算简单概率

When all outcomes are equally likely, the probability of an event can be found using a simple formula:

当所有结果等可能时，一个事件的概率可以通过一个简单的公式求得：

Probability = Number of favourable outcomes / Total number of outcomes

For instance, if you roll a fair six-sided dice, the total number of outcomes is 6. The number of favourable outcomes for rolling an even number (2, 4 or 6) is 3. Therefore, the probability of rolling an even number is 3/6, which simplifies to 1/2.

例如，如果你掷一颗均匀的六面骰子，总结果数为 6。掷出偶数（2、4 或 6）的有利结果数是 3。因此，掷出偶数的概率是 3/6，约简后为 1/2。

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5. Sample Space | 样本空间

The sample space is the set of all possible outcomes of an experiment. Listing the sample space clearly helps to avoid missing or double-counting outcomes.

样本空间是一个实验所有可能结果的集合。清晰地列出样本空间有助于避免遗漏或重复计算结果。

For example, when flipping two coins, the sample space can be written as {HH, HT, TH, TT}, where H stands for heads and T for tails. From this, you can see there are 4 equally likely outcomes, and the probability of getting at least one head is 3/4.

例如，抛掷两枚硬币时，样本空间可以写作 {正正, 正反, 反正, 反反}，其中“正”表示正面，“反”表示反面。由此可以看出有 4 种等可能的结果，而至少出现一个正面的概率是 3/4。

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6. Equally Likely Outcomes | 等可能结果

The basic probability formula only works when all outcomes are equally likely. This means each outcome has the same chance of occurring as any other.

基本的概率公式仅在所有结果等可能时才适用。这意味着每一个结果发生的可能性与其他结果完全相同。

Throwing a fair dice or flipping a fair coin produces equally likely outcomes. However, if a dice is biased, some faces may appear more often, so the outcomes are no longer equally likely and the simple formula cannot be used without adjustments.

掷一颗均匀的骰子或抛一枚均匀硬币会产生等可能的结果。然而，如果骰子有偏斜，某些面出现的频率可能更高，这样各结果就不再等可能，简单公式也无法直接使用。

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7. Probability as a Fraction, Decimal, or Percentage | 概率表示为分数、小数或百分比

Probability can be expressed in three equivalent forms: fractions, decimals and percentages. A probability of 1/2 can also be written as 0.5 or 50%.

概率可以用三种等价形式表示：分数、小数和百分比。概率 1/2 也可以写作 0.5 或 50%。

When solving problems, you may need to convert between these forms. For example, the probability 3/5 equals 0.6 and 60%. Using a percentage can often make comparisons clearer.

在解题时，你可能需要在这些形式之间进行转换。例如，概率 3/5 等于 0.6 和 60%。使用百分比常常能使比较更为清晰。

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

Theoretical probability is calculated using known possible outcomes and the assumption of equal likelihood. Experimental probability, on the other hand, comes from carrying out an experiment or survey.

理论概率是利用已知的可能结果以及等可能的假设计算出来的。另一方面，实验概率来自于进行实验或调查。

For example, the theoretical probability of rolling a 3 on a dice is 1/6. If you roll the dice 60 times and get a 3 on 12 occasions, the experimental probability is 12/60 = 1/5. The more trials you conduct, the closer the experimental probability tends to get to the theoretical probability.

例如，掷骰子得到 3 点的理论概率是 1/6。如果你掷骰子 60 次，出现 3 点有 12 次，那么实验概率就是 12/60 = 1/5。你进行的试验次数越多，实验概率往往越接近理论概率。

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9. Expected Frequency | 期望频数

You can predict how many times an event is likely to occur by calculating the expected frequency. This is found by multiplying the probability of the event by the number of trials.

通过计算期望频数，你可以预测某个事件大概会发生多少次。计算方法是用事件的概率乘以试验次数。

Expected Frequency = Probability x Number of Trials

If the probability of rain on a given day is 0.3, then over 50 days you would expect it to rain on 0.3 x 50 = 15 days. Remember that this is only an expectation; actual results can vary.

如果某一天下雨的概率是 0.3，那么在 50 天中，你预计会有 0.3 x 50 = 15 天下雨。请记住，这只是一个期望值，实际结果可能会有出入。

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10. Mutually Exclusive Events | 互斥事件

Two events are mutually exclusive if they cannot happen at the same time. For a single dice roll, getting a 2 and getting a 5 are mutually exclusive because both outcomes cannot occur together.

如果两个事件不可能同时发生，则称它们是互斥事件。就单次掷骰子而言，得到 2 点和得到 5 点是互斥的，因为两个结果不可能一同出现。

If A and B are mutually exclusive, the probability of either A or B occurring is the sum of their individual probabilities: P(A or B) = P(A) + P(B).

如果 A 与 B 互斥，那么发生 A 或 B 的概率是它们各自概率的和：P(A 或 B) = P(A) + P(B)。

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11. The Complement of an Event | 事件的补集

The complement of an event A is the event ‘not A’, which includes all outcomes not in A. The probabilities of an event and its complement always add up to 1.

事件 A 的补集是指“非 A”事件，它包含了所有不在 A 中的结果。一个事件与其补集的概率之和总是等于 1。

P(not A) = 1 – P(A)

If the probability of winning a game is 0.2, then the probability of not winning is 0.8. Using complements often simplifies calculations when finding the probability of an event by considering what should not happen.

如果赢得一场游戏的概率是 0.2，那么赢不了的概率就是 0.8。在求概率时，利用补集常常能通过考虑不应发生的情况来简化计算。

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12. Probability Rules and the Sum of All Outcomes | 概率规则与所有结果之和

The sum of the probabilities of all mutually exclusive outcomes in a sample space is always 1. This rule helps to check that a set of probabilities is valid.

样本空间中所有互斥结果的概率之和总是为 1。这条规则有助于验证一组概率是否合理。

For two mutually exclusive events A and B, we can write: P(A) + P(B) + … = 1 for all possible disjoint outcomes. When events are not mutually exclusive, the addition rule becomes P(A or B) = P(A) + P(B) – P(A and B), but at KS3 you will mainly focus on mutually exclusive events.

对于两个互斥事件 A 和 B，我们可以写出所有不相交的可能结果满足 P(A) + P(B) + … = 1。当事件并非互斥时，加法法则变为 P(A 或 B) = P(A) + P(B) – P(A 且 B)，但在 KS3 阶段你主要聚焦于互斥事件。

Understanding these basic rules allows you to solve a wide variety of probability problems, from simple games to more challenging puzzles involving combined events.

理解这些基本法则后，你就能解决各种各样的概率问题，从简单的游戏到涉及组合事件的更具挑战性的谜题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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