Conditional Probability
Introduced| English | Meaning |
|---|---|
| conditional probability/kənˈdɪʃənl ˌprɒbəˈbɪlɪti/ | The probability of an event given that another event has occurred. |
| multiplication rule/ˌmʌltɪplɪˈkeɪʃn ruːl/ | P(A and B) = P(A) × P(B | A). |
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."
On a tree diagram, multiply the probabilities along the branches
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.
Write from a conditional probability of 0.3
- 50 students are in a club.
- 15 of them play piano. The lesson conditional probability 0.4 is another group.
What is the probability a student plays piano, given the club?
Conditional probability divides by the size of the given group.
What is the probability of piano given the club?
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15 divided by 50 is 0.3. Other options — 0.4 (the other group); 15 (the piano count).
Conditional probabilities
How the chance of B changes depending on whether A happened.
P(A and B) = 0.2 and P(B) = 0.5. Find P(A | B).
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P(A|B) = P(A∩B)/P(B) = 0.2/0.5 = 0.4.
In general, P(A | B) equals P(B | A).
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They are usually different — the condition matters.
The general multiplication rule for P(A and B) is...
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P(A∩B) = P(B)·P(A|B), chaining the two events.
To read P(A | B) off a two-way table, you divide by the...
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The condition B becomes the denominator — its margin total.
P(A | B) is read 'the probability of A ___ B' (one word).
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The bar means 'given' the condition.