SL 4.11—Formal conditional probability

Conditional probability restricts the denominator.

P(A|B)=P(A∩B)/P(B) describes A inside the cases where B has occurred, so the reference population changes.

Example

Among 100 tested devices, 20 are defective and 8 of those are overheating: P(overheating|defective)=8/20, not 8/100.

Write the conditioning event under the denominator and use a two-way table when possible.

P(A|B) and P(B|A) answer different questions.

Equivalent multiplication form: $P(A\cap B)=P(B)P(A|B)$. Events are independent exactly when conditioning does not change the probability, for example $P(A|B)=P(A)$ (with $P(B)>0$); the syllabus also permits comparison with $P(A|B')$. A numerical equality must be shown rather than assuming independence from the wording.