📚 Review set 19A - Non-Calculator: Mastering Probability | 概率复习集 19A - 无计算器解题技巧
Review set 19A in the IB Mathematics course is designed to test your understanding of fundamental probability concepts without the aid of a calculator. This set focuses on logical reasoning, basic counting, fraction arithmetic, and the ability to apply probability laws mentally. In this article, we break down the key ideas behind the non-calculator questions so you can approach them with confidence.
IB 数学课程中的复习题集 19A 旨在不使用计算器的情况下考查您对基本概率概念的理解。题目侧重于逻辑推理、简单计数、分数运算以及心算概率定律的能力。本文将剖析这些无计算器题型背后的核心思想，帮助您自信应对。
1. Understanding the No-Calculator Mindset | 理解无计算器解题思路
When solving probability problems without a calculator, you must rely on fraction simplification, mental arithmetic, and a clear understanding of probability rules. The questions are designed so that numbers are manageable, often with small denominators. Always simplify fractions and leave answers in exact form, such as 1/3, 2/5, or 7/12. Avoid decimals unless explicitly requested.
在无计算器解决概率问题时，您需要依靠分数化简、心算以及对概率规则清晰的理解。题目设计的数字一般易于处理，分母较小。需始终化简分数，并以精确形式（如 1/3、2/5 或 7/12）作答。除非明确要求，否则避免使用小数。
For example, if you roll a fair six‑ 2. No calculator is needed to simplify this fraction or to add probabilities like P(even) + P(prime) – P(even and prime).
例如，掷一个公平六面骰子，得到偶数的概率为 P(偶数) = 3/6 = 1/2。化简这个分数或计算像 P(偶数) + P(质数) – P(偶数且质数) 这类概率相加都不需要计算器。
2. Core Probability Axioms and Notation | 核心概率公理与符号
le space S satisfies P(S) = 1, and an impossible event has probability 0. The notation P(A ∪ B) represents the union (A or B), P(A ∩ B) the intersection (A and B), and P(A') or P(Ā ng these symbols helps you translate word problems into algebraic probability expressions.
回想一下，对于任何事件 A，均有 0 ≤ P(A) ≤ 1。样本空间 S 的概率为 P(S) = 1，不可能事件的概率为 0。符号 P(A ∪ B) 表示并集（A 或 B），P(A ∩ B) 表示交集（A 且 B），P(A') 或 P(Ā) 表示 A 的补集。掌握这些符号有助于将文字问题转化为代数概率表达式。
In many non‑calculator items, you will be given probabilities such as P(A) = 1/3, P(B) = 1/4, and P(A ∩ 2. You can then apply rules to find other probabilities without reaching for a calculator.
在许多无计算器题目中，您会得到诸如 P(A) = 1/3、P(B) = 1/4 及 P(A ∩ B) = 1/12 这样的概率，然后运用定律求出其他概率，完全无需计算器。
3. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法定律
Two events are mutually exclusive if they cannot occur at the same time, so P(A ∩ B) = 0. In that case, the addition rule simplifies to P(A ∪ nts from the context, such as drawing a heart or a spade from a standard deck in one draw.
若两事件不可能同时发生，则为互斥事件，此时 P(A ∩ B) = 0。加法定律简化为 P(A ∪ B) = P(A) + P(B)。您必须能从上下文中识别互斥事件，例如从一副标准扑克牌中一次抽牌，抽到红心或黑桃。
If they are not mutually exclusive, remember the general addition formula:
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
若事件不互斥，请牢记一般加法公式：
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
robability that a randomly chosen student studies Mathematics or Physics is (18/30 + 15/30 – 10/30) = 23/30 – all manageable by mental arithmetic.
例如，一个 30 名学生的班级中，18 人学数学，15 人学物理，10 人两门都学。随机选一名学生，他学习数学或物理的概率为 (18/30 + 15/30 – 10/30) = 23/30 —— 全部可心算完成。
4. Complementary Events | 互补事件
The complement of event A, denoted A', consists of all outcomes not in A. By definition, P(A) + P(A') = 1, so P(A') = 1 – P(A). This simple relationship is extremely useful in non‑ problems, especially when it is easier to find the probability of something not happening.
事件 A 的补集 A' 包含所有不属于 A 的结果。根据定义，P(A) + P(A') = 1，因此 P(A') = 1 – P(A)。这一简单关系在无计算器问题中极为有用，尤其是当更容易求出某个事件不发生的概率时。
If the probability of raining on a given day is 2/5, the probability of no rain is 1 – ay, you can quickly compute complements by subtracting the numerator from the denominator.
如果某天下雨的概率为 2/5，那么不下雨的概率就是 1 – 2/5 = 3/5。在分数情形下，只需用分母减分子即可快速算出补集。
5. Independent Events and the Multiplication Rule | 独立事件与乘法定律
ccurrence of one does not affect the probability of the other. For independent events, P(A ∩ B) = P(A) × P(B). In a non‑ mple fractions like (1/6) × (1/2) = 1/12, which is straightforward with mental arithmetic.
两事件 A 与 B 相互独立，若其中一个发生不影响另一个的概率。对独立事件而言，P(A ∩ B) = P(A) × P(B)。在无计算器环境中，经常要计算诸如 (1/6) × (1/2) = 1/12 的简单分数乘法，用心算就很容易。
t (e.g., two rolls of a fair die). Be prepared to test for independence by checking if P(A ∩ B) = P(A) × P(B) when given certain probabilities.
但不要想当然地假定独立，除非题目明确说明或背景隐含（例如两次掷一个公平骰子）。当给定某些概率时，要准备好通过验证 P(A ∩ B) 是否等于 P(A) × P(B) 来检验独立性。
6. Conditional Probability Made Simple | 条件概率简化
Conditional probability P(A|B) means the probability of A occurring given that B has occurred. The formula is:
P(A|B) = P(A ∩ B) / P(B)
条件概率 P(A|B) 表示已知 B 发生的条件下 A 发生的概率。公式为：
P(A|B) = P(A ∩ B) / P(B)
Non‑calculator questions often provide fractions that divide nicely. For example, if P(A ∩ B) = 1/8 and P(B) = 1/4, then P(A| /4) = (1/8) × (4/1) = 1/2. Practice fraction division without a calculator to build speed.
无计算器题目通常给出的分数能整齐相除。例如，若 P(A ∩ B) = 1/8，P(B) = 1/4，则 P(A|B) = (1/8) ÷ (1/4) = (1/8) × (4/1) = 1/2。请多练习分数除法以提高速度。
7. Tree Diagrams without Technology | 不依赖技术的树形图
Tree diagrams are an excellent tool for organising sequential events. Without a calculator, you can draw a simple tree and label branches with probabilities as fractions. Then multiply along the branches to find compound probabilities and add the relevant branch probabilities for ‘at least one’ scenarios.
树形图是理清连续事件的绝佳工具。无计算器时，您可以画出简单树状图，用分数标注各条分支概率。然后沿分支相乘得到复合概率，再通过相加相关分支概率来解决“至少一个”的情形。
For instance, a bag contains 4 red and 6 blue counters. Two counters are drawn without replacement. The tree will have branches 4/10, 6/10, then 3/9, 5/9, etc. The arithmetic remains fraction‑based, so a calculator is unnecessary.
例如，一个袋子里有 4 个红色和 6 个蓝色筹码，无放回地抽取两次。树形图分支标注 4/10、6/10，然后 3/9、5/9 等。整个运算只涉及分数，因而无需计算器。
8. Venn Diagrams and Set Probability | 韦恩图与集合概率
Venn diagrams help visualise unions, intersections, and complements. In a typical non‑ ions. Always remember that P(A) = n(A) / n(S), where n(S) is the total number of outcomes.
韦恩图有助于直观展示并集、交集和补集。在典型的无计算器题目中，您可能会得到韦恩图各区域的元素个数，并被要求以分数形式求概率。始终牢记 P(A) = n(A) / n(S)，其中 n(S) 为结果总数。
agram and find that the probability a randomly selected student likes neither sport is (50 – (16+12+8)) / 50 = 14/50 = 7/25. All numbers are designed for mental calculation.
考虑一项 50 名学生的调查：28 人喜欢足球，20 人喜欢篮球，12 人两者都喜欢。您可以填充韦恩图得出，随机选一名学生两种运动都不喜欢的概率为 (50 – (16+12+8)) / 50 = 14/50 = 7/25。所有数字都是为心算设计的。
9. Counting Principles and Simple Combinatorics | 计数原理与简单组合
Some non‑ × 1) = 10. You can handle these without a calculator by simplifying factorial expressions.
有些无计算器题目需要利用乘法原理或组合来计数。例如，将 3 本不同的书排列在书架上，共有 3 × 2 × 1 = 6 种方式。若要从 5 人中选出一个 2 人委员会，组合数为 (5 × 4) / (2 × 1) = 10。您可以通过化简阶乘表达式来处理这些问题，无需计算器。
When calculating probabilities with equally likely outcomes, break down numerator and denominator using the fundamental counting principle. Reduce fractions by cancelling common factors before multiplying, which makes the arithmetic even simpler.
在计算等可能结果的概率时，使用基本计数原理分解分子和分母。相乘前通过约分来化简分数，这样算术运算会更简单。
10. Tackling Typical Non-Calculator Questions | 攻克典型无计算器题目
has 2³ = 8 equally likely outcomes. The outcomes with exactly two heads are HHT, HTH, THH – 3 outcomes. So P(exactly two heads) = 3/8. No calculator needed.
我们来看一道改编自复习集 19A 风格的经典问题：一枚公平硬币抛掷三次，求恰好得到两次正面的概率。样本空间共有 2³ = 8 种等可能结果。恰好两次正面的结果为 HHT、HTH、THH，共 3 种。因此 P(恰好两正面) = 3/8。完全不需要计算器。
Another common type: Given P(A) = 0.4, P(B) = 0.5, and P(A ∪ B) = 0.7, find P(A ∩ B). Rearranging the addition formula gives P(A ∩ B) = 0.4 + 0.5 – 0.7 = 0.2 = 1/5. Although decimals are used, the arithmetic is elementary.
另一种常见类型：已知 P(A) = 0.4, P(B) = 0.5, P(A ∪ B) = 0.7，求 P(A ∩ B)。重新整理加法公式得 P(A ∩ B) = 0.4 + 0.5 – 0.7 = 0.2 = 1/5。尽管使用了小数，但运算十分基础。
11. Common Pitfalls and How to Avoid Them | 常见陷阱与避免方法
Many students lose marks by not simplifying fractions, misapplying independence, or confusing P(A|B) with P(B|A). Always double‑check whether events are independent or not, and never round probabilities unless instructed. In non‑calculator tasks, exact fractions are expected.
许多学生因为没有化简分数、错误套用独立性或混淆 P(A|B) 与 P(B|A) 而失分。务必仔细核对事件是否独立，且除非有明确要求，否则不要四舍五入概率值。在无计算器题目中，要求使用精确分数。
- Mistake: Using P(A ∪ B) = P(A) + P(B) for non‑mutually exclusive events. Fix: Subtract P(A ∩ B).
- Mistake: Assuming independence without checking. Fix: Verify P(A ∩ B) = P(A) × P(B).
- Mistake: Forgetting to reduce fractions, e.g., leaving 4/16 instead of 1/4.
- 错误：对非互斥事件使用 P(A ∪ B) = P(A) + P(B)。纠正：减去 P(A ∩ B)。
- 错误：未经检验就假定独立。纠正：验证 P(A ∩ B) = P(A) × P(B)。
- 错误：忘记约分，例如留下 4/16 而非 1/4。
12. Practice Strategy for Review Set 19A | 复习集 19A 的练习策略
To excel in non‑calculator probability questions, practice with simple integers and fractions regularly. Work through all parts of Review set 19A, simulating exam conditions: no calculator, neat working, and all steps shown. Rehearse fraction operations (addition, subtraction, multiplication, division) until they become automatic.
要出色完成无计算器概率题目，应定期用简单的整数和分数进行练习。模拟考试环境完成复习集 19A 的所有部分：不使用计算器，书写整洁，步骤完整。反复演练分数四则运算，直到变得自动化。
Create your own mini‑problems: roll dice, flip coins, draw cards, and compute probabilities manually. The more comfortable you are with manipulating fractions without technology, the more efficiently you will handle Review set 19A and similar assessments.
自编小型问题：掷骰子、抛硬币、抽纸牌，并手动计算概率。您对无技术辅助的分数运算越自如，应对复习集 19A 及类似评估的效率就越高。
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