Set Operations

Set Operations are the fundamental ways of combining or modifying sets — union, intersection, complement, and difference — which translate directly into logical statements about events in probability.

$A \cup B = \{\, \omega : \omega \in A \text{ or } \omega \in B \,\}$

$A \cap B = \{\, \omega : \omega \in A \text{ and } \omega \in B \,\}$

$A^c = S \setminus A, \qquad A \setminus B = A \cap B^c$

Union means "or", intersection means "and", the complement $A^c$ means "not $A$ ", and the difference $A \setminus B$ means "in $A$ but not $B$ ".
De Morgan's laws relate them: $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$ .
On disjoint events $A \cap B = \varnothing$ , so $P(A \cup B) = P(A) + P(B)$ .

Finding $P(A^c \cap B)$ from Given Probabilities If $P(A) = 0.4$ , $P(B) = 0.5$ , and $P(A \cap B) = 0.2$ , what is $P(A^c \cap B)$ ?

Answer $A^c \cap B$ is the part of $B$ that does not overlap $A$ : $P(A^c \cap B) = P(B) - P(A \cap B) = 0.5 - 0.2 = 0.3$