条件概率Conditional Probability
已知一件事发生后，应该在哪个范围内重新计算比例？Once an event is known to have occurred, which population should the proportion use?
前置知识：Prerequisites: B12：事件、概率与独立性。B12: events, probability and independence.
复习前置：Review prerequisites: B12 · 随机事件、概率与独立性Random Events, Probability & Independence
学完后你能：After this lesson you can: 选择条件范围，计算条件比例，并识别零分母。Select the conditioned population, compute a conditional proportion, and recognize a zero denominator.
动手观察Explore
选择“已知 B”或“已知 A”，观察保留的格子、分子和分母；选择不可能条件查看未定义状态。Select “given B” or “given A” to inspect retained cells, numerator and denominator; select an impossible condition to see the undefined state.
从观察到解释From observation to explanation
先缩小范围First narrow the population
在 100 个对象中，A 且 B 有 18 个，非 A 且 B 有 12 个。因此已知 B 时，只保留 30 个对象，其中 A 有 18 个：P(A|B)=18/30=0.6。Among 100 objects, 18 have both A and B, and 12 have B without A. Given B, retain only 30 objects; 18 have A, so P(A|B)=18/30=0.6.
交换条件Swap the condition
若 A 且非 B 有 42 个，那么 A 总数为 60。已知 A 时，P(B|A)=18/60=0.3。分子都是交集，分母却不同，因此两个条件概率一般不同。If 42 have A without B, there are 60 A objects. Given A, P(B|A)=18/60=0.3. The numerator is the same intersection, but the denominators differ, so the conditional probabilities generally differ.
还原联合概率Recover the joint probability
P(A∩B)=P(B)P(A|B)。本例中 P(B)=0.3，因此 0.3×0.6=0.18，与 18/100 一致。P(A∩B)=P(B)P(A|B). Here P(B)=0.3, so 0.3×0.6=0.18, matching 18/100.
公式与符号Formula and symbols
A、B 是事件；竖线表示“已知 B”；A∩B 表示两者都发生。P(B) 是筛选后范围所占比例，必须大于零。A and B are events; the vertical bar means “given B,” and A∩B means both occur. P(B) is the probability of the filtered population and must be positive.
留意这个区别Watch for this
条件事件概率为零时，上面的比值没有定义；不要把它显示成 0。P(A|B) 也不能随意与 P(B|A) 互换。When the conditioning event has probability zero, the ratio is undefined, not zero. P(A|B) and P(B|A) cannot generally be swapped.
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