Conditional Probability · 条件概率
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| conditional probability/kənˈdɪʃənl ˌprɒbəˈbɪlɪti/ | 条件概率 | tiáo jiàn gài lǜ |
| multiplication rule/ˌmʌltɪplɪˈkeɪʃn ruːl/ | 乘法法则 | chéng fǎ fǎ zé |
Probability, given a clue
- A conditional probability 条件概率 is the chance of $A$ given that $B$ has happened.
- Written $P(A \mid B)$ — "the probability of $A$ given $B$."
- Knowing $B$ occurred shrinks the world to just the $B$ outcomes.
- Then you ask what fraction of those are also $A$.
已知线索下的概率
- 条件概率是在已知 $B$ 已发生的情况下 $A$ 的机会。
- 记作 $P(A \mid B)$——“在 $B$ 条件下 $A$ 的概率”。
- 知道 $B$ 发生了，就把世界缩小到只剩 $B$ 的那些结果。
- 然后你问那些当中有多少同时也是 $A$。
The formula
-
$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$
- The numerator is "both happen"; the denominator is "the condition."
- You're rescaling: out of the $B$-world, how much is also $A$?
- The condition $B$ becomes the new "total."
公式
-
$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$
- 分子是“两者都发生”；分母是“那个条件”。
- 你在重新标度：在 $B$ 的世界里，有多少也是 $A$？
- 条件 $B$ 成了新的“总数”。
From a two-way table
- Two-way tables make conditionals easy: restrict to the condition's row or column.
- $P(A \mid B)$ = (cell for both) ÷ (total of $B$'s row/column).
- This is exactly the conditional relative frequency from Unit 2.
- The denominator is a margin, not the grand total.
从双向表
- 双向表让条件概率变得容易：限制到条件所在的行或列。
- $P(A \mid B)$ =（两者都的单元格）÷（$B$ 的行/列合计）。
- 这正是第 2 单元里的条件相对频率。
- 分母是一个边际，而不是总计数。
The general multiplication rule
- Rearranging the formula gives the multiplication rule 乘法法则:
-
$$P(A \cap B) = P(B)\,P(A \mid B)$$
- "Both happen" = (first happens) × (second happens given the first).
- It's how you chain probabilities of events that depend on each other.
一般乘法法则
- 把公式变形就得到乘法法则：
-
$$P(A \cap B) = P(B)\,P(A \mid B)$$
- “两者都发生” =（第一个发生）×（在第一个条件下第二个发生）。
- 这就是你如何把相互依赖的事件的概率串起来。
$P(A \mid B)$ and $P(B \mid A)$ are usually different — don't swap them. "Probability of a cough given the flu" is high; "probability of the flu given a cough" is low (most coughs aren't flu). The condition — what's given — sits after the bar and becomes the denominator. Read carefully which event is the condition.
$P(A \mid B)$ 和 $P(B \mid A)$ 通常不同——不要互换它们。“在患流感条件下咳嗽的概率”很高；“在咳嗽条件下患流感的概率”很低（大多数咳嗽不是流感）。条件——被给定的东西——位于竖线之后，并成为分母。仔细读清哪个事件是条件。
In a class, $P(\text{plays sport}) = 0.5$ and $P(\text{sport and music}) = 0.2$.
- $P(\text{music} \mid \text{sport}) = \dfrac{0.2}{0.5} = 0.4$.
- Among the sporty half, $40\%$ also do music.
- Multiplication check: $P(\text{sport and music}) = 0.5 \times 0.4 = 0.2$. ✓
在一个班里，$P(\text{plays sport}) = 0.5$，$P(\text{sport and music}) = 0.2$。
- $P(\text{music} \mid \text{sport}) = \dfrac{0.2}{0.5} = 0.4$。
- 在爱运动的那一半人中，$40\%$ 也搞音乐。
- 乘法验证：$P(\text{sport and music}) = 0.5 \times 0.4 = 0.2$。✓
A conditional probability $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ is the chance of $A$ given $B$ — restrict to $B$'s row/column in a two-way table. Rearranged, the multiplication rule $P(A \cap B) = P(B)\,P(A \mid B)$. Remember $P(A\mid B) \ne P(B \mid A)$ in general.
条件概率 $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ 是在给定 $B$ 下 $A$ 的机会——在双向表中限制到 $B$ 的行/列。变形后得到乘法法则 $P(A \cap B) = P(B)\,P(A \mid B)$。记住一般情况下 $P(A\mid B) \ne P(B \mid A)$。
Conditional probabilities · 条件概率
How the chance of B changes depending on whether A happened. · B 的机会如何随 A 是否发生而改变。
P(A and B) = 0.2 and P(B) = 0.5. Find P(A | B). · P(A 且 B) = 0.2，P(B) = 0.5。求 P(A | B)。
P(A|B) = P(A∩B)/P(B) = 0.2/0.5 = 0.4. · P(A|B) = P(A∩B)/P(B) = 0.2/0.5 = 0.4。
In general, P(A | B) equals P(B | A). · 一般来说，P(A | B) 等于 P(B | A)。
They are usually different — the condition matters. · 它们通常不同——条件很重要。
The general multiplication rule for P(A and B) is... · P(A 且 B) 的一般乘法法则是……
P(A∩B) = P(B)·P(A|B), chaining the two events. · P(A∩B) = P(B)·P(A|B)，把两个事件串联。
To read P(A | B) off a two-way table, you divide by the... · 要从双向表读出 P(A | B)，你要除以……
The condition B becomes the denominator — its margin total. · 条件 B 成为分母——它的边际合计。
P(A | B) is read 'the probability of A ___ B' (one word). · P(A | B) 读作“在 B ___ 下 A 的概率”（填英文一词 given）。
The bar means 'given' the condition. · 竖线表示“在……条件下（given）”。