📚 IB Math: Introduction to Probability | IB数学：概率初步
Probability is the branch of mathematics that quantifies uncertainty. From predicting the outcome of a dice roll to analysing real-world data, probability forms the foundation of statistics and is a core topic in the IB Mathematics curriculum.
概率是数学中量化不确定性的分支。从预测掷骰子的结果到分析现实世界的数据，概率构成了统计学的基础，也是IB数学课程中的核心主题。
1. What is Probability? | 什么是概率？
Probability measures the likelihood that a particular event will occur. It is always a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain.
概率衡量某一特定事件发生的可能性。它的取值始终在0和1之间，其中0表示事件不可能发生，1表示事件必然发生。
For example, the probability of rolling a 3 on a fair six-sided die is 1/6, while the probability of rolling a 7 is 0, and the probability of rolling a number less than 7 is 1.
例如，掷一枚均匀六面骰子得到3的概率是1/6，而得到7的概率是0，得到小于7的数的概率是1。
2. Sample Space and Events | 样本空间与事件
The sample space S is the set of all possible outcomes of an experiment. An event A is any subset of the sample space.
样本空间S是实验中所有可能结果组成的集合。事件A是样本空间的任意子集。
Consider rolling a fair six-sided die. The sample space is S = {1, 2, 3, 4, 5, 6}. If we define event A as “rolling an even number”, then A = {2, 4, 6}.
考虑掷一枚均匀六面骰子。样本空间为S = {1, 2, 3, 4, 5, 6}。如果我们定义事件A为“掷出偶数”，那么A = {2, 4, 6}。
In IB problems, you should first clearly identify the sample space. For a finite sample space with equally likely outcomes, the probability of an event A is given by:
在IB题目中，你应首先明确识别样本空间。对于结果等可能的有限样本空间，事件A的概率为：
P(A) = n(A) / n(S)
where n(A) is the number of outcomes in A and n(S) is the total number of outcomes in S.
其中n(A)是事件A包含的结果数，n(S)是样本空间S中结果的总数。
3. Basic Probability Rules | 基本概率规则
The probability of any event satisfies the following fundamental rules:
任何事件的概率都满足以下基本规则：
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0 ≤ P(A) ≤ 1 for every event A.
对于任意事件A，有0 ≤ P(A) ≤ 1。
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P(S) = 1; the probability of the entire sample space is 1.
P(S) = 1；整个样本空间的概率为1。
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If A and B are events, then P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
若A和B是事件，则P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
The addition rule above applies to all events, whether or not they overlap. The term P(A ∩ B) is subtracted because outcomes in both A and B are counted twice.
上述加法公式适用于所有事件，无论它们是否重叠。减去P(A ∩ B)是因为同时属于A和B的结果被重复计算了两次。
4. Complementary Events | 对立事件
The complement of an event A, written A′, consists of all outcomes in the sample space that are not in A. The complement rule states:
事件A的对立事件，记为A′，由样本空间中所有不属于A的结果组成。对立事件公式为：
P(A′) = 1 − P(A)
This rule is especially useful when calculating the probability of “at least one” occurrence, which is often easier to compute as 1 minus the probability of “none”.
这个公式在计算“至少一次”发生的概率时尤为有用，通常转化为1减去“一次也不发生”的概率来简化计算。
For example, if the probability that it rains on a given day is 0.3, then the probability that it does not rain is 0.7.
例如，若某天下雨的概率是0.3，则不下雨的概率是0.7。
5. Probability of Combined Events | 复合事件的概率
For two events A and B, the probability of their union is given by the addition rule:
对于两个事件A和B，它们并集的概率由加法公式给出：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Here, A ∪ B represents the event that A or B (or both) occurs, and A ∩ B represents the event that both A and B occur.
这里A ∪ B表示A或B（或两者同时）发生的事件，A ∩ B表示A和B同时发生的事件。
If the events are mutually exclusive, then A ∩ B = ∅, so the formula simplifies to P(A ∪ B) = P(A) + P(B).
如果事件互斥，则A ∩ B = ∅，因此公式简化为P(A ∪ B) = P(A) + P(B)。
6. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot occur at the same time. In set notation, A ∩ B = ∅, and therefore P(A ∩ B) = 0.
如果两个事件不能同时发生，则称它们为互斥事件。用集合记号表示，即A ∩ B = ∅，因此P(A ∩ B) = 0。
For mutually exclusive events, the addition rule becomes:
对于互斥事件，加法公式变为：
P(A ∪ B) = P(A) + P(B)
A common example is rolling a die and considering the events “rolling a 2” and “rolling a 5”. These events cannot happen simultaneously, so their union probability is 1/6 + 1/6 = 1/3.
一个常见例子是掷骰子并考虑“掷出2”和“掷出5”两个事件。这两个事件不可能同时发生，因此它们的并集概率是1/6 + 1/6 = 1/3。
7. Independent Events and Conditional Probability | 独立事件与条件概率
Two events are independent if the occurrence of one does not affect the probability of the other. For independent events A and B:
如果事件A的发生不影响事件B发生的概率，则称这两个事件相互独立。对于独立事件A和B：
P(A ∩ B) = P(A) × P(B)
Conditional probability, written as P(A|B), is the probability of A occurring given that B has already occurred. It is defined by:
条件概率记为P(A|B)，表示在B已经发生的条件下A发生的概率。其定义为：
P(A|B) = P(A ∩ B) / P(B)
For independent events, P(A|B) = P(A), because the condition B provides no extra information about A.
对于独立事件，P(A|B) = P(A)，因为条件B不提供关于A的额外信息。
| Relationship | Condition | Key formula |
| Mutually exclusive | A ∩ B = ∅ | P(A ∪ B) = P(A) + P(B) |
| Independent | P(A|B) = P(A) | P(A ∩ B) = P(A) × P(B) |
Note that mutually exclusive and independent are two different concepts. Mutually exclusive events cannot be independent unless one of them has probability zero.
注意互斥和独立是两个不同的概念。除非其中一个事件的概率为零，否则互斥事件不可能是独立的。
8. Tree Diagrams and Venn Diagrams | 树状图与韦恩图
Tree diagrams are useful for multi-stage experiments. Each branch represents a possible outcome, and probabilities on the branches are multiplied to find the probability of a path.
树状图适用于多阶段实验。每条分支代表一个可能结果，将分支上的概率相乘即可得到某条路径的概率。
For example, drawing two balls from a bag with replacement can be represented on a tree. The probability of a particular sequence is the product of the probabilities along the branches.
例如，从袋中有放回地抽取两个球，可以用树状图表示。某个特定序列的概率等于各分支概率的乘积。
Venn diagrams display the relationships between events. The overlapping region represents A ∩ B, and the entire shaded region represents A ∪ B. They are especially helpful when applying the addition rule.
韦恩图展示事件之间的关系。重叠区域表示A ∩ B，整个阴影区域表示A ∪ B。在运用加法公式时，韦恩图尤其有用。
9. Probability with Counting Principles | 计数原理与概率
When outcomes are equally likely, counting techniques such as permutations and combinations can be used to compute probabilities.
当结果等可能时，可以使用排列与组合等计数技巧来计算概率。
The number of ways to choose r objects from n distinct objects (without regard to order) is:
从n个不同的对象中选取r个对象（不考虑顺序）的方法数为：
C(n, r) = n! / [r! (n − r)!]
For instance, the probability of drawing two red cards from a standard deck without replacement can be found by counting favourable combinations divided by the total combinations.
例如，从一副标准扑克牌中不放回地抽取两张红牌的概率，可以通过有利组合数除以总组合数来求得。
10. Common Exam Pitfalls | 常见考试陷阱
Many IB students lose marks by confusing mutually exclusive and independent events, or by forgetting whether an experiment is with or without replacement.
许多IB学生在互斥事件与独立事件之间产生混淆，或忘记实验是否有放回，从而失分。
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Check whether the problem says “with replacement” or “without replacement”. For without replacement, the denominator decreases after each draw.
检查题目是否写明“有放回”或“无放回”。若是无放回，每次抽取后分母会减少。
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Do not use P(A ∩ B) = P(A) × P(B) unless independence is guaranteed.
除非题目明确保证独立性，否则不要使用P(A ∩ B) = P(A) × P(B)。
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For “at least one” problems, use the complement rule P(at least one) = 1 − P(none).
对于“至少一次”问题，使用对立事件公式P(至少一次) = 1 − P(一次都没有)。
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Always state the sample space clearly in simple probability questions.
在简单概率题中务必清楚写出
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