📚 Probability: Sample Spaces, Mutually Exclusive Events and Sum of Probabilities | 概率：样本空间、互斥事件与概率之和
This revision guide explains the Cambridge KS3 probability topic often tested in Checkpoint. You will learn how to describe likelihood, list all possible outcomes in a sample space, use the key probability rules correctly, and avoid the common errors that cost marks. Worked examples, exam tips and practice questions with answers are included to help you build confidence and accuracy.
本复习指南讲解剑桥 KS3 概率专题中 Checkpoint 常考的内容。你将学习如何描述事件发生的可能性、列出样本空间中的所有可能结果、正确运用核心概率规则，并避免常见的失分错误。文中配有例题、应考技巧以及带答案的练习题，帮助你建立信心与准确性。
1. Probability Scale and Key Terms | 概率标度与关键术语
The probability scale runs from 0 to 1. A probability of 0 means an event is impossible, and a probability of 1 means it is certain. A value of 1/2 describes an event that is equally likely to happen or not happen, such as getting heads on a fair coin. Values closer to 1 are more likely; values closer to 0 are less likely.
概率标度从 0 到 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。数值 1/2 描述发生与不发生可能性相同的事件，例如抛一枚均匀硬币得到正面。数值越接近 1 越可能发生，越接近 0 越不可能发生。
ecimals or percentages. Converting between them is a frequent requirement: 1/4 = 0.25 = 25%, and 3/5 = 0.6 = 60% wer in the form the question asks for, and simplify fractions fully unless told otherwise.
概率可以写成分数、小数或百分数，三者之间的转换是常见要求：1/4 = 0.25 = 25%，3/5 = 0.6 = 60%。最终答案应使用题目要求的形式，除题目另有说明外，分数要化成最简形式。
0 ≤ P(E) ≤ 1
The expression above states that the probability of any event E always lies between 0 and 1 inclusive. If you calculate a negative probability, or a probability greater than 1, you have definitely made a mistake somewhere.
上面的式子表示：任何事件 E 的概率永远位于 0 与 1 之间（含端点）。如果你算出负概率，或大于 1 的概率，那一定是在某处出错了。
| Term 术语 | Meaning 含义 | Example 示例 |
|---|---|---|
| Impossible 不可能 | Probability 0 | Rolling a 7 on a normal 6-sided dice |
| Unlikely 不大可能 | Probability closer to 0 | Rolling a 6 (probability 1/6) |
| Equally likely 等可能 | Same probability | Getting heads or tails on a fair coin |
| Likely 很可能 | Probability closer to 1 | Rolling an even number (probability 1/2) |
| Certain 必然 | Probability 1 | The sun rising tomorrow |
When a question uses words such as "fair", "unbiased" or "at random", it is telling you that every listed outcome is equally likely. That single word is the signal to use the basic definition: probability equals the number of favourable outcomes divided by the total number of outcomes.
当题目中出现 fair（均匀）、unbiased（无偏）或 at random（随机）等词时，它是在告诉你所列的每个结果都是等可能的。这个词就是提示你使用基本定义：概率等于有利结果数除以结果总数。
P(E) = number of favourable outcomes ÷ total number of outcomes
2. Sample Space Diagrams | 样本空间图
A sample space is the set of all possible outcomes of an experiment. For a single event, the sample space is simply a list: rolling a fair dice gives {1, 2, 3, 4, 5, 6}. For two combined events, a sample space diagram is a table showing every possible combination. This helps you count outcomes systematically and avoid missing or double-counting any result.
样本空间是一次试验中所有可能结果的集合。对于单一事件，样本空间就是一张列表：掷一枚均匀骰子得到 {1, 2, 3, 4, 5, 6}。对于两个组合事件，样本空间图是一张显示所有可能组合的表格，帮助你系统计数，避免遗漏或重复计数。
To draw a sample space diagram, put the outcomes of one event along the top and the outcomes of the other event down the side. Every cell inside the grid represents one combined outcome, written as an ordered pair such as (2, 3) or as a label such as 2H.
画样本空间图时，把一个事件的结果写在上方一行，把另一个事件的结果写在左侧一列。表格中的每一格代表一个组合结果，可以写成有序对，例如 (2, 3)，也可以写成标签，例如 2H。
There are 3 × 3 = 9 equally likely outcomes, and the totals are written inside the cells.
下表显示两个均匀三面转盘 A 和 B（分别标有 1、2、3）的样本空间。共有 3 × 3 = 9 个等可能结果，格内写的是两数之和。
| A ↓ / B → | 1 | 2 | 3 |
|---|---|---|---|
| 1 | 2 | 3 | 4 |
| 2 | 3 | 4 | 5 |
| 3 | 4 | 5 | 6 |
2 appears once, so P(total = 2) = 1/9, and the total 1 never appears, so it is impossible.
从这张图可以直接读出概率。例如，和为 4 出现在三格中，所以 P(和为 4) = 3/9 = 1/3。和为 2 只出现一次，所以 P(和为 2) = 1/9，而和为 1 从不出现，因此是不可能事件。
3. Counting Outcomes Systematically | 系统地计数结果
Listing outcomes carelessly is the main reason students lose marks in probability. Always work in a fixed order: fix the first outcome and run through all the possibilities of the second, then fix the next first outcome, and so on. This "fixed first, vary second" method guarantees that nothing is missed and nothing is repeated.
随意列举结果是学生在概率题上失分的主要原因。请始终按固定顺序进行：先固定第一个结果，依次列出第二个结果的所有可能；再固定下一个第一个结果，依此类推。这种"固定前者、变化后者"的方法能保证既不遗漏也不重复。
n be made in m ways and an independent second choice in n ways, the total number of combined outcomes is m × n. Three stages multiply together in the same way.
当选项只有两三个时，直接写出完整列表又快又稳。数量较大时，可用计数原理：如果第一个选择有 m 种方式，独立的第二个选择有 n 种方式，那么组合结果总数就是 m × n。三个阶段的情况也以同样方式连乘。
Total outcomes = m × n × p × …
inations. If a set meal also includes one of 3 desserts, the total becomes 4 × 6 × 3 = 72.
例如，一家咖啡店有 4 种三明治和 6 种饮料，那么共有 4 × 6 = 24 种午餐组合。如果套餐还包含 3 种甜点中的一种，总数就变成 4 × 6 × 3 = 72 种。
cribes. If the same item cannot be chosen twice, such as picking two different students from a group, the count becomes m × (m − 1) rather than m × m.
请记住，计数原理只在各选择真正独立、且顺序符合题意时才适用。如果同一项不能被选两次，例如从一组学生中选两名不同的学生，那么数量就变成 m × (m − 1)，而不是 m × m。
4. The Sum of Probabilities for All Outcomes | 所有结果的概率之和
If you list every possible outcome of an experiment, the probabilities of those outcomes must add up to exactly 1. This is because one of the outcomes is certain to happen, and the outcomes do not overlap. This rule is one of the most useful tools in the whole topic.
如果你列出一次试验的所有可能结果，那么这些结果的概率之和必然恰好等于 1。原因是这些结果中必定有一个会发生，而且它们彼此不重叠。这条法则是整个专题中最实用的工具之一。
P(outcome 1) + P(outcome 2) + … + P(outcome n) = 1
lue) = 0.35 and P(green) = 0.3. Then P(yellow) = 1 − (0.2 + 0.35 + 0.3) = 1 − 0.85 = 0.15.
这条法则让你用减法求缺失的概率。假设一个偏心转盘可以停在红、蓝、绿或黄，已知 P(红) = 0.2、P(蓝) = 0.35、P(绿) = 0.3，那么 P(黄) = 1 − (0.2 + 0.35 + 0.3) = 1 − 0.85 = 0.15。
) = 1 − 7/12 = 5/12. Always use a common denominator when adding fractions, and simplify your final answer.
同样的思路也适用于分数。如果袋中只有红、蓝、绿三种棋子，且 P(红) = 1/4、P(蓝) = 1/3，那么 P(绿) = 1 − (1/4 + 1/3) = 1 − 7/12 = 5/12。分数相加时务必先通分，最后把结果化成最简形式。
A quick check at the end of any probability calculation is to add up all your probabilities and see whether they total 1. This habit catches arithmetic slips before they cost you marks.
在任何概率计算结束时，快速把所有概率相加，看看是否等于 1。这个习惯能在算术失误真正丢分之前就把它抓住。
5. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. Rolling an odd number and rolling an even number on a single dice are mutually exclusive, because a single roll cannot be both. In a Venn diagram, mutually exclusive events are shown as two separate circles that do not overlap.
如果两个事件不可能同时发生，就说它们是互斥的。掷一枚骰子得到奇数与得到偶数就是互斥的，因为一次投掷不可能既是奇数又是偶数。在文氏图中，互斥事件画成两个互不重叠的圆。
Many pairs of events are not mutually exclusive. Rolling an even number and rolling a multiple of 3 are not mutually exclusive, because rolling a 6 satisfies both conditions at the same time. Any overlapping outcome means the events are not mutually exclusive.
很多事件对并不是互斥的。掷出偶数与掷出 3 的倍数就不互斥，因为掷出 6 同时满足两个条件。只要存在共同的结果，两个事件就不是互斥的。
| Pair of events 事件对 | Mutually exclusive? 是否互斥 | Reason 原因 |
|---|---|---|
| Rolling a 2 / rolling a 5 | Yes | One roll cannot be both 2 and 5 |
| Rolling a prime / rolling a 4 | Yes | 4 is not prime, so no overlap |
| Rolling an even number / rolling a prime | No | 2 is both even and prime |
| Rolling a number less than 3 / rolling a number greater than 4 | Yes | No number satisfies both |
A related idea is exhaustive events: a set of events is exhaustive if together they cover every possible outcome. For a single dice, the events "rolling a 1", "rolling a 2", …, "rolling a 6" are both mutually exclusive and exhaustive, which is exactly why their probabilities sum to 1.
与之相关的概念是完备事件组：如果一组事件合起来覆盖了所有可能结果，就称它们是完备的。对于一枚骰子，事件"掷出 1""掷出 2"……"掷出 6"既互斥又完备，这正是它们的概率之和为 1 的原因。
6. The Addition Rule for "Or" | "或"的加法法则
When two events are mutually exclusive, the probability that one or the other happens is found by adding their probabilities. This is called the addition rule, and it applies only when there is no overlap between the events.
当两个事件互斥时，"其中一个发生"的概率就是它们概率之和。这称为加法法则，只适用于两事件没有重叠的情形。
P(A or B) = P(A) + P(B) (A and B mutually exclusive)
= 2/6 = 1/3. Equally, P(rolling a number greater than 4) = P(5) + P(6) = 1/6 + 1/6 = 1/3.
例如，掷一次均匀骰子，P(掷出 1 或 2) = 1/6 + 1/6 = 2/6 = 1/3。同样，P(掷出大于 4 的数) = P(5) + P(6) = 1/6 + 1/6 = 1/3。
If the events overlap, adding the probabilities counts the overlap twice, so you must subtract it once. This is the general addition rule, and it explains why in the dice example P(even or prime) is not simply 3/6 + 3/6.
如果两个事件有重叠，直接相加会把重叠部分算两次，因此必须减去一次。这就是一般的加法法则，它也解释了为什么在上面的骰子例子中，P(偶数或质数)并不等于 3/6 + 3/6。
P(A or B) = P(A) + P(B) − P(A and B)
= 3/6 + 3/6 − 1/6 = 5/6. Checking the outcomes directly, 2, 3, 4, 5 and 6 all qualify, which is 5 out of 6.
代入正确数字：对于均匀骰子，P(偶数) = 3/6，P(质数) = 3/6（质数为 2、3、5），而 P(偶数且质数) = P(2) = 1/6。所以 P(偶数或质数) = 3/6 + 3/6 − 1/6 = 5/6。直接核对结果：2、3、4、5、6 全部符合，即 6 个中的 5 个。
7. Complementary Events | 对立事件
The complement of an event A is the event that A does not happen, written A′. Because an event either happens or does not happen, these two outcomes are mutually exclusive and exhaustive, so their probabilities add to 1.
事件 A 的对立事件是"A 不发生"，记作 A′。因为一个事件要么发生要么不发生，这两种情形互斥且完备，所以它们的概率之和为 1。
P(A′) = 1 − P(A)
The complement rule is especially powerful for questions containing the phrase "at least" omplement is instant: P(at least one head) = 1 − P(no heads) = 1 − (1/2)³ = 1 − 1/8 = 7/8.
对立事件法则对含有"至少"（at least）的题目尤其有力。例如，抛一枚均匀硬币三次，如果逐一列举来求"至少出现一次正面"的概率会很耗时，但用对立事件则立刻可得：P(至少一次正面) = 1 − P(没有正面) = 1 − (1/2)³ = 1 − 1/8 = 7/8。
The rule works with percentages too. If the probability that it rains tomorrow is 35%, the probability that it does not rain is 100% − 35% = 65% hatever form you use, the two probabilities must always combine to make exactly one whole.
这条法则对百分数同样适用。如果明天下雨的概率是 35%，那么不下雨的概率就是 100% − 35% = 65%。无论用哪种形式，两个概率相加都必须正好等于 1 个整体。
A useful sanity check is that a complement is always less than or equal to 1 and greater than or equal to 0. If you obtain P(A′) as a negative value, then P(A) must have been greater than 1, which signals an earlier error.
一个实用的检查是：对立事件的概率永远在 0 到 1 之间（含端点）。如果你算出 P(A′) 为负值，那么 P(A) 一定曾大于 1，这说明前面出现了错误。
8. Experimental Probability and Relative Frequency | 实验概率与相对频率
Theoretical probability is calculated from the structure of the situation, such as 1/6 for each face of a fair dice. Experimental probability, also called relative frequency, is calculated from the results of an actual experiment: you divide the number of times the event occurred by the total number of trials.
理论概率根据情境的结构计算，例如均匀骰子每一面的概率是 1/6。实验概率又称相对频率，是根据实际实验的结果计算的：用事件发生的次数除以试验总次数。
Relative frequency = number of successful trials ÷ total number of trials
l probability of point down is 130/200 = 0.65. Notice that the two relative frequencies still add to 1.
假设一枚图钉被抛落 200 次，其中钉尖朝上 70 次，那么钉尖朝上的实验概率是 70/200 = 0.35，朝下的实验概率是 130/200 = 0.65。注意这两个相对频率依然相加等于 1。
Relative frequency is not a fixed value: repeat the experiment and you will usually get a slightly different result. However, as the number of trials increases, the relative frequency tends to get closer to the theoretical probability. This tendency is why large samples give more reliable estimates.
相对频率并不是固定值：重复实验通常会得到略有不同的结果。但是随着试验次数增加，相对频率会趋近于理论概率。这一趋势正是大样本给出的估计更可靠的原因。
Exam questions often ask you to use relative frequency to estimate how many times an event will occur in a future experiment, or to judge whether a dice or spinner is biased. If an observed relative frequency stays a long way from the expected theoretical value after many trials, that is evidence of bias.
考题常要求你用相对频率去估计未来实验中事件会发生的次数，或判断骰子、转盘是否有偏。如果在大量试验之后，观察到的相对频率仍然远离理论值，那就是存在偏差的证据。
9. Expected Number of Occurrences | 期望发生次数
Once you know a probability, you can predict how many times an event is expected to happen in a given number of trials. Multiply the probability of the event by the number of trials. The answer is an expectation, not a guarantee, so it may be a non-whole number.
一旦知道概率，就能预测在给定试验次数中事件预计会发生多少次：用事件的概率乘以试验次数。得到的是期望值，而不是保证值，因此可能是非整数。
Expected number = P(event) × number of trials
50 = 45. If a fair coin is tossed 80 times, the expected number of heads is 0.5 × 80 = 40.
例如，一枚有偏骰子 P(掷出 6) = 0.18，掷 250 次，则期望出现 6 的次数为 0.18 × 250 = 45 次。如果抛一枚均匀硬币 80 次，期望正面的次数为 0.5 × 80 = 40 次。
If a question expects a whole-number answer, round sensibly and state that your figure is an estimate. For instance, an expected value of 37.5 would usually be reported as approximately 38 times in context.
如果题目要求整数答案，请合理取整，
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