📚 Probability and Sample Spaces | 概率与样本空间
Probability is the branch of mathematics that measures how likely an event is to happen. In Cambridge KS3 mathematics, you learn to describe chance on a scale from 0 to 1, list all possible outcomes, and use diagrams such as sample spaces and tree diagrams to solve problems. This article reviews the key ideas and common exam techniques for probability.
概率是数学中衡量某一事件发生可能性的分支。在剑桥 KS3 数学中，你将学习用 0 到 1 之间的数值描述可能性，列出所有可能结果，并利用样本空间和树状图等工具解决问题。本文复习概率的核心概念和常见考试技巧。
1. The Probability Scale | 概率标尺
Probability is always written as a number between 0 and 1 inclusive. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain. Values closer to 0 are less likely, while values closer to 1 are more likely.
概率总是用 0 到 1 之间的数表示，包括 0 和 1。概率为 0 表示事件不可能发生，概率为 1 表示事件必然发生。越接近 0 表示越不可能，越接近 1 表示越有可能。
You can express probability as a fraction, decimal, or percentage. For example, a probability of 1/4 can also be written as 0.25 or 25%.
概率可以用分数、小数或百分数表示。例如，概率 1/4 也可以写成 0.25 或 25%。
To calculate the probability of an event when all outcomes are equally likely, use the formula below.
当所有结果等可能时，可用以下公式计算事件的概率。
P(event) = number of favourable outcomes ÷ total number of outcomes
This formula only works when every outcome has the same chance of occurring.
这个公式仅在所有结果发生的机会均等时成立。
2. Listing Outcomes and Sample Spaces | 列出结果与样本空间
The sample space is the set of all possible outcomes of an experiment. For example, when you roll a fair six-sided dice, the sample space is {1, 2, 3, 4, 5, 6}. Listing outcomes systematically helps you count accurately and avoid missing or repeating outcomes.
样本空间是一次试验所有可能结果的集合。例如，掷一枚公平的六面骰子时，样本空间为 {1, 2, 3, 4, 5, 6}。系统列出结果有助于准确计数，避免遗漏或重复。
ordered pairs such as (H, 1), (H, 2), ..., (T, 6). There are 2 × 6 = 12 possible outcomes.
对于两个事件，可以用样本空间图表示所有组合。如果同时抛一枚硬币和掷一枚骰子，结果可以写成 (H, 1)、(H, 2)、……、(T, 6) 这样的有序对。共有 2 × 6 = 12 种可能结果。
of combined outcomes is m × n. This is known as the multiplication principle for counting.
当两个独立事件分别有 m 和 n 种可能结果时，组合结果的总数为 m × n。这就是计数的乘法原理。
3. Two-Way Tables | 双向表
A two-way table is a useful way to display outcomes involving two categories, such as colour and size. Each cell shows the frequency or count for one combination. Totals in the margins help you check that the table is complete.
双向表是展示两个类别结果的有用工具，例如颜色和尺寸。每个单元格显示某一组合的频数或计数。边栏的总数有助于检查表格是否完整。
tal. For example, if 7 out of 20 counters are red and large, then P(red and large) = 7/20.
从双向表求概率时，先找出符合条件的单元格或总数，再除以总计数。例如，若 20 个计数器中红色且大的有 7 个，则 P(红色且大) = 7/20。
Always check that the row totals and column totals add up to the grand total before calculating probabilities.
在计算概率之前，务必检查行总数和列总数是否相加等于总计数。
4. Sample Space Diagrams for Two Dice | 两个骰子的样本空间图
. These can be displayed in a 6 by 6 grid showing all ordered pairs from (1, 1) to (6, 6).
掷两枚公平骰子时，共有 6 × 6 = 36 种等可能结果。可以用一个 6×6 的网格显示从 (1, 1) 到 (6, 6) 的所有有序对。
To find the probability of a particular sum, count the number of ordered pairs that give that sum. For example, the sum of 7 occurs with the pairs (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1), so there are 6 favourable outcomes.
要求出某一特定点数和数的概率，先数出能得到该和数的有序对个数。例如，点数和为 7 的有序对有 (1, 6)、(2, 5)、(3, 4)、(4, 3)、(5, 2) 和 (6, 1)，因此共有 6 个有利结果。
P(sum = 7) = 6/36 = 1/6
This shows that a sum of 7 is the most likely total when two fair dice are rolled.
这表明掷两枚公平骰子时，点数和为 7 是最可能出现的总点数。
5. 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 dice, the events 'rolling a 2' and 'rolling a 5' are mutually exclusive because a single roll cannot show both numbers.
如果两个事件不能同时发生，则称它们为互斥事件。例如，掷骰子时，“掷出 2 点”和“掷出 5 点”是互斥事件，因为一次掷骰不可能同时出现两个点数。
For mutually exclusive events A and B, the probability that either A or B occurs is the sum of their individual probabilities.
对于互斥事件 A 和 B，事件 A 或 B 发生的概率等于两者各自概率之和。
P(A or B) = P(A) + P(B)
This is called the addition rule for mutually exclusive events. It only works when the events do not overlap.
这称为互斥事件的加法法则。仅当事件没有重叠时才适用。
6. Complementary Events | 互补事件
of A. If the probability of rain is 0.3, then the probability of no rain is 1 - 0.3 = 0.7.
事件 A 的互补事件是指 A 不发生的事件。互补事件的概率等于 1 减去 A 的概率。如果下雨的概率是 0.3，则不下雨的概率为 1 - 0.3 = 0.7。
P(not A) = 1 - P(A)
This rule is very helpful when it is easier to find the probability of the opposite event. For example, finding the probability of getting at least one head in two coin flips is quicker by subtracting the probability of no heads from 1.
当求相反事件的概率更容易时，这一法则非常有用。例如，求抛两枚硬币至少出现一次正面的概率时，用 1 减去没有正面的概率会更快。
7. Tree Diagrams for Independent Events | 独立事件的树状图
Tree diagrams show the possible outcomes of two or more independent events. Along each branch, write the probability of that outcome. To find the probability of a combination of outcomes, multiply the probabilities along the relevant branches.
树状图展示两个或多个独立事件的可能结果。在每条分支上标出该结果的概率。要求某一结果组合的概率，只需将相应分支上的概率相乘。
second counter is drawn. The probability of drawing red then blue is (3/5) × (2/5) = 6/25.
例如，一个袋子中有 3 个红色和 2 个蓝色计数器。抽出一个后放回，再抽第二个。先抽红色再抽蓝色的概率为 (3/5) × (2/5) = 6/25。
Because the first counter is replaced, the probabilities on the second draw stay the same as on the first draw. This is the key feature of independent events.
因为第一个
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