条件付き確率

The detective's question

• You know someone studies French. What's the chance they also study German?
• This isn't the same as "what's the chance someone studies both?" — you've narrowed the group to just the French students.
• That's conditional probability 条件概率: the chance of $B$ given that $A$ has already happened.

How to read it

• $\text{P}(B \mid A)$ means "the probability of $B$ given $A$".
• Look only at the part that matches the condition — on a Venn diagram, two-way table 双向表, or tree.

On a tree diagram, multiply probabilities along the branches; here two reds give $\tfrac38\times\tfrac38=\tfrac{9}{64}$

The formula (Extended)

On a two-way table

• To find $\text{P}(B \mid A)$: look at the row (or column) for $A$, and find the fraction that is also $B$.
• The row total is the denominator, not the grand total.

Independent vs dependent 相依的

• If $\text{P}(B \mid A) = \text{P}(B)$, the events are independent — knowing $A$ happened doesn't change the chance of $B$.
• If they differ, the events are dependent.

Conditional probability restricts attention to the group matching the condition — like looking only inside one region of a Venn diagram.

Condition on the smaller group

• Of 30 students, 18 study French and 7 of these study German. Given French, use $p=7/18$, not $7/30$; the condition selects only 18 students.
• Suppose 12 students in the full class study German. Then $P(G)=12/30=0.4$ differs from $P(G\mid F)=7/18$. The two events are not independent for these data.

You've got it