| English | 의미 |
|---|---|
| conditional probability/kənˈdɪʃənl ˌprɒbəˈbɪlɪti/ | 조건부 확률 |
| multiplication rule/ˌmʌltɪplɪˈkeɪʃn ruːl/ | 곱의 법칙 |
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 $A$.
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."
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.
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.
$P(A \mid B)$ 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.
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: $P(\text{sport and music}) = 0.5 \times 0.4 = 0.2$. ✓
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.
조건부 확률들
A가 일어났는지 여부에 따라 B의 확률이 어떻게 변하는지
P(A且B) = 0.2이고 P(B) = 0.5일 때 P(A | B)를 구하십시오.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
P(A|B) = P(A∩B)/P(B) = 0.2/0.5 = 0.4.
일반적으로 P(A | B)는 P(B | A)와 같습니다.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
보통 다릅니다 — 조건이 중요합니다.
P(A且B)에 대한 일반 곱셈 법칙은...
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
P(A∩B) = P(B)·P(A|B), 두 사건을 연결하여 계산합니다.
이원 분포표에서 P(A | B)를 읽으려면 ...로 나누어야 합니다.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
조건 B가 분모가 됩니다 — 가장자리 합계입니다.
P(A | B)는 'A ___ B의 확률'로 읽습니다 (단어 하나).
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
막대 기호는 '조건이 주어졌을 때'를 의미합니다.