📚 GCSE OCR Maths: Probability Key Points | 概率 考点精讲
Probability is a fundamental topic in GCSE OCR Mathematics, requiring a clear understanding of chance, sample spaces, and event relationships. This guide walks you through every key concept, from the basic probability scale to tree diagrams, conditional probability and Venn diagrams, all tailored to the OCR specification. Each section provides bilingual explanations and examples to help you master exam-style questions and avoid common pitfalls.
概率是 GCSE OCR 数学中的核心考点，要求清晰理解可能性、样本空间和事件关系。本指南带你逐一攻破所有重点，从基础的概率尺度到树形图、条件概率和维恩图，完全贴合 OCR 考试大纲。每个小节都配有中英双语讲解和示例，帮助你掌握考试题型，避开常见失分点。
1. Probability Scale and Basics | 概率尺度与基础
Probability is a measure of how likely an event is to occur. It can be expressed as a fraction, decimal, or percentage and always lies between 0 and 1 inclusive. A probability of 0 means the event is impossible, while a probability of 1 means it is certain.
概率是事件发生可能性的度量。它可以用分数、小数或百分比表示，并且总是在 0 和 1 之间（包含两端）。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。
ery likely can be mapped to numerical ranges. For any event A, the probability that A does not occur is 1 minus the probability that it does occur: P(not A) = 1 - P(A).
概率尺度有助于直观理解可能性：0 表示不可能，0.5 表示机会均等，1 表示必然发生。不太可能、可能、非常可能等词语可以映射到数值范围。对于任何事件 A，不发生 A 的概率等于 1 减去发生 A 的概率：P(非 A) = 1 - P(A)。
2. Sample Spaces and Outcomes | 样本空间与结果
A sample space is the set of all possible outcomes of an experiment. For example, tossing a fair coin gives the sample space {Head, Tail}. Rolling a fair six-sided die gives {1, 2, 3, 4, 5, 6}. To work out probabilities accurately, you must first identify the sample space and ensure all outcomes are equally likely.
样本空间是一次试验所有可能结果的集合。例如，抛一枚均匀硬币的样本空间为 {正面, 反面}。掷一个均匀的六面骰子的样本空间为 {1, 2, 3, 4, 5, 6}。要想准确计算概率，首先要确定样本空间，并确保每个结果发生的可能性相等。
For two or more combined events, you can list the sample space systematically using a table or a grid. For instance, when rolling two dice, the sample space has 36 outcomes. A two-way table helps to count favourable outcomes. Writing the sample space as a set of ordered pairs prevents double-counting or omissions.
对于两个或更多组合事件，可以用表格或网格系统地列出样本空间。例如，掷两颗骰子时，样本空间有 36 种结果。双向表有助于计数有利结果的数量。将样本空间写为有序数对可以防止重复计数或遗漏。
3. Basic Probability Formula | 基本概率公式
If all outcomes are equally likely, the probability of an event is given by the formula:
P(Event) = Number of favourable outcomes / Total number of outcomes
如果所有结果等可能发生，事件的概率由下式给出：
P(事件) = 有利结果的数量 / 总结果的数量
olling a prime number on a fair die is 3/6 = 1/2 because the prime numbers are 2, 3 and 5.
这常写作 P(A) = n(A) / n(S)，其中 A 是我们关心的事件，S 是样本空间。例如，掷一个均匀骰子得到质数的概率是 3/6 = 1/2，因为质数有 2、3 和 5。
Always check that your fraction is simplified and that probabilities are given in the form asked for in the exam — typically a fraction in its simplest form, a decimal, or a percentage.
一定要检查分数是否已经约简，并且概率是否以试卷要求的格式给出——通常是最简分数、小数或百分比。
4. Mutually Exclusive Events and the OR Rule | 互斥事件与“或”规则
Two events are mutually exclusive if they cannot happen at the same time. For example, getting a Head and getting a Tail on a single coin toss are mutually exclusive. When events are mutually exclusive, the probability that either event A or event B occurs is the sum of their individual probabilities:
P(A or B) = P(A) + P(B)
如果两个事件不能同时发生，则它们互斥。例如，一次抛硬币得到正面与得到反面是互斥事件。当事件互斥时，事件 A 或事件 B 发生的概率等于各自概率之和：
P(A 或 B) = P(A) + P(B)
If the events are not mutually exclusive, you must subtract the probability that both occur to avoid double-counting:
P(A or B) = P(A) + P(B) - P(A and B)
如果事件不互斥，则必须减去两事件同时发生的概率，以避免重复计数：
P(A 或 B) = P(A) + P(B) - P(A 且 B)
In OCR questions, you will often need to decide whether events are mutually exclusive, and then apply the appropriate rule. Drawing a Venn diagram or listing outcomes can help you decide.
在 OCR 考题中，你需要判断事件是否互斥，然后应用相应的公式。画维恩图或列出结果有助于做出判断。
5. Independent Events and the AND Rule | 独立事件与“且”规则
Events are independent if the outcome of one does not affect the probability of the other occurring. For example, flipping a coin and rolling a die are independent events. When events A and B are independent, the probability that both occur is found by multiplying their probabilities:
P(A and B) = P(A) × P(B)
如果一个事件的结果不影响另一个事件发生的概率，则它们是独立事件。例如，抛硬币和掷骰子是独立事件。当 A 与 B 独立时，两者同时发生的概率通过相乘得到：
P(A 且 B) = P(A) × P(B)
Be careful: the multiplication rule for independent events only applies when events are truly independent. If the question involves selecting items without replacement, the events are not independent — the probability changes after the first selection. In such cases, tree diagrams with conditional probabilities are the safest approach.
请注意：独立事件的乘法规则仅在事件确实独立时才适用。如果题目涉及不放回抽取，事件就不独立——第一次选择后概率会改变。在这种情况下，使用带有条件概率的树形图是最稳妥的方法。
6. Tree Diagrams | 树形图
Tree diagrams are a powerful tool for showing all possible outcomes of two or more events, especially when events are not independent. Each branch represents a possible outcome, and you write the probability along the branch. The probabilities from each fork must sum to 1.
树形图是展示两个或更多事件所有可能结果的有力工具，特别适用于事件不独立的情况。每条分支代表一种可能结果，概率标在分支上。每个分叉处的各分支概率之和必须为 1。
To find the probability of a combination of outcomes, multiply the probabilities along the path. If more than one path leads to the same final event, add the path probabilities together. This is the 'multiply along branches, add across branches' principle.
要找出某个组合结果的概率，将沿该路径的分支概率相乘。如果不止一条路径通向同一个最终事件，则将各路径的概率相加。这就是“沿分支相乘，跨分支相加”的原则。
When drawing a tree diagram for selection without replacement, remember that the total number of items decreases and the probabilities on the second set of branches change accordingly. For conditional probability questions, you can often place the relevant conditional probabilities directly on the second layer of branches.
绘制不放回抽取的树形图时，要记住总物品数会减少，第二层分支上的概率会相应改变。对于条件概率问题，通常可以将相关的条件概率直接标注在第二层分支上。
7. Conditional Probability | 条件概率
Conditional probability is the probability of an event B occurring given that event A has already occurred, written as P(B|A). It is especially important in OCR higher tier questions and can be calculated using the formula:
P(B|A) = P(A and B) / P(A)
条件概率是指已知事件 A 已发生的情况下，事件 B 发生的概率，记作 P(B|A)。它在 OCR 高等级题目中尤为重要，可通过以下公式计算：
P(B|A) = P(A 且 B) / P(A)
This formula makes sense if you think of restricting the sample space to only those outcomes where A happens. Tree diagrams often have conditional probabilities built into the second set of branches. If a question states that an item is selected and not replaced, you are dealing with conditional probability.
如果将样本空间限制在仅发生 A 的那些结果上，这个公式就很有意义。树形图的第二层分支往往就包含了条件概率。如果题目说明物品被选出且不放回，那么你处理的就是条件概率。
In exam problems, you might be asked to work out P(A and B) from a two-way table or a Venn diagram and then use it to find a conditional probability. Always check that the denominator is the probability of the given event, not the whole sample space.
在考试题目中，可能会要求你从双向表或维恩图求出 P(A 且 B)，然后用它来计算条件概率。务必检查分母是已知事件的概率，而不是整个样本空间的概率。
8. Venn Diagrams and Set Notation | 维恩图与集合符号
Venn diagrams represent events as overlapping circles inside a rectangle that stands for the sample space. The intersection A ∩ B is the overlap region and represents outcomes where both A and B occur. The union A ∪ B includes all outcomes that are in A or B or both.
维恩图用大矩形（样本空间）内重叠的圆圈来表示事件。交集 A ∩ B 是重叠区域，表示 A 和 B 同时发生的结果。并集 A ∪ B 包含属于 A 或属于 B 或同时属于两者的所有结果。
Probabilities can be written inside the regions or derived from given numbers. P(A) is the total probability in circle A. P(A ∩ B) is the probability in the intersection. The complement of A, written A', covers everything outside A. Knowing set notation is essential for OCR exam questions that ask for probabilities in terms of ∩ and ∪.
概率可以写在各个区域内，或者根据给定的数字推导得出。P(A) 是圆圈 A 内的总概率。P(A ∩ B) 是交集内的概率。A 的补集，记作 A'，包含 A 之外的所有结果。掌握集合符号对 OCR 考试中要求用 ∩ 和 ∪ 表示概率的题目至关重要。
To find P(A ∪ B) using a Venn diagram, you can add the probabilities in circles A and B, then subtract the intersection probability if it has been counted twice. This matches the general OR rule.
使用维恩图求 P(A ∪ B) 时，可以将圆圈 A 和 B 中的概率相加，然后减去被重复计算一次的交集概率。这与一般的“或”规则相吻合。
9. Expected Frequency | 期望频率
If an experiment is repeated many times, you can predict how often an event will occur using the formula:
Expected frequency = Probability of event × Number of trials
如果一个试验重复多次，你可以用下面的公式预测事件发生的次数：
期望频率 = 事件的概率 × 试验次数
u will get exactly 50 sixes, but over many trials the average tends towards the expected value.
例如，如果你掷一个均匀骰子 300 次，出现六点的期望次数是 (1/6) × 300 = 50。这并不意味着你一定会恰好得到 50 次六点，但经过大量试验后，平均值会趋近于期望值。
Expected frequency questions often appear alongside relative frequency and can test your understanding of theoretical versus experimental probability. Always show the multiplication step clearly.
期望频率的题目常与相对频率一同出现，可以考查你对理论概率与实验概率的理解。务必清晰地展示乘法步骤。
10. Relative Frequency and Experimental Probability | 相对频率与实验概率
Relative frequency is an estimate of probability based on experimental data. It is calculated as:
Relative frequency = Number of times event occurs / Total number of trials
相对频率是基于实验数据对概率的估计。其计算方法为：
相对频率 = 事件发生的次数 / 试验总次数
As the number of trials increases, the relative frequency usually gets closer to the theoretical probability. This is known as the law of large numbers. OCR questions may ask you to calculate a relative frequency and then comment on whether the die or coin appears to be fair.
随着试验次数增加，相对频率通常会越来越接近理论概率，这被称为大数定律。OCR 考题可能会要求你计算相对频率，然后对骰子或硬币是否均匀做出评述。
When comparing experimental and theoretical probabilities, remember that variation is expected in small samples. You might use a calculated relative frequency to find an expected number of future successes in a new set of trials.
当比较实验概率与理论概率时，要记住小样本中预计会出现差异。你可以用计算出的相对频率来预测新一轮试验中成功的期望次数。
11. Probability of 'At Least One' and Complementary Events | “至少一次”与互补事件
When calculating the probability of 'at least one' success, it is often much easier to work out the probability of the complementary event — that none succeed — and subtract from 1. This approach avoids having to sum many separate probabilities.
当计算“至少一次”成功的概率时，通常先求出互补事件——即一次都不成功的概率，再将其从 1 中减去，这种方法要简便得多。它避免了繁琐地加总多个独立的概率。
P(at least one) = 1 - P(none)
P(至少一次) = 1 - P(全不成功)
For example, if the probability that a bias-free thrower hits the target is 0.3, then the probability of at least one hit in three throws is 1 - (0.7)3. This method works well with tree diagrams, especially when all trials are independent and identical.
例如，若一名无偏差的投手命中目标的概率为 0.3，那么三次投掷中至少命中一次的概率为 1 - (0.7)3。这种方法在树形图中尤其好用，特别是当各次试验独立且完全相同时。
Key reminder: the complement of 'at least one' is 'none', not 'less than one'. Always verify that the events you are using as complements truly cover all possibilities and are mutually exclusive.
重要提醒：“至少一次”的互补事件是“一次都不”，而不是“少于一次”。务必确认你用作互补的事件确实覆盖了所有可能性并且互斥。
12. Exam Tips and Common Pitfalls | 考试技巧与常见错误
Draw diagrams whenever possible — a quick tree diagram or Venn sketch can transform a confusing problem into a clear set of paths or regions. Label probabilities carefully and check that branch probabilities from a single point sum to 1.
尽可能画图——一个简单的树形图或维恩草图可以将令人困惑的问题转化为清晰的路径或区域。仔细标注概率，并检查同一点分出的各分支概率之和是否为 1。
Watch out for replacement: without replacement, the probabilities on second branches depend on the first outcome, so do not blindly use the AND rule for independent events. Always ask yourself whether the events are independent or conditional.
注意有放回还是无放回：在不放回情况下，第二分支的概率取决于第一次的结果，因此切勿盲目使用独立事件的“且”规则。始终问问自己事件是独立事件还是条件事件。
Do not confuse mutually exclusive with independent. Mutually exclusive events cannot happen together, while independent events have no influence on each other's probability. Mistaking one for the other is a very common error.
不要混淆互斥与独立。互斥事件不能同时发生，而独立事件彼此的概率不受对方影响。混淆二者是非常常见的错误。
Keep your fractions simplified and present your final answer in the form requested. If the question asks for a percentage, convert your decimal or fraction appropriately. Even a correct method can lose a mark if the final answer is not in the specified form.
保持分数最简，并按照题目要求的格式给出最终答案。如果要求以百分比作答，须正确转换小数或分数。即便解题方法正确，若最终答案格式不符合要求，仍可能丢分。
Finally, always read the question carefully to determine the exact event whose probability is needed — words like 'given that', 'at least', and 'or' are crucial clues to the structure of the calculation.
最后，始终仔细审题，明确需要计算哪个事件的概率——“已知”、“至少”、“或”这类词语是决定计算结构的关键线索。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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