SL 4.6—Combined, conditional and independent events

Combine events by matching the rule to their relationship.

Use addition for mutually exclusive alternatives, multiplication for a sequence, and conditional probability when the first event changes the sample space.

Example

If P(A)=0.4 and P(B|A)=0.5, then P(A∩B)=0.2; independence would require P(B|A)=P(B).

Draw a tree or table and label whether each branch is conditional or independent.

P(A∩B)=P(A)P(B) only when independence is justified, not merely because events are different.

General addition rule: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$; only mutually exclusive events have $P(A\cap B)=0$. Conditional probability is $P(A|B)=P(A\cap B)/P(B)$, so $P(A\cap B)=P(B)P(A|B)$. Independence requires $P(A\cap B)=P(A)P(B)$. Without replacement normally changes later branch probabilities; with replacement can preserve them.