We are given events A, B. If "the probability of at least one of A and B happens" translates to Pr($A$ ∪ $B$), and "at most one of A and B happens" is the complement of the former, would the probability be Pr($A$ ∪ $B$)c?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
The probability that at most one of $A$ or $B$ happens can be broken down into 3 disjoint cases: neither $A$ nor $B$ happens, $A$ but not $B$ happens, and $B$ but not $A$ happens. This is: $$Pr[(A \cup B)^c \cup (A \cap B^c) \cup (A^c \cap B)]$$ $$=Pr[(A \cap B^c) \cup (A^c \cap B^c) \cup (A^c \cap B)]$$ $$=Pr[(A \cup A^c) \cap B^c \cup A^c \cap(B^c \cup B)]$$ $$=Pr[B^c \cup A^c]$$ $$=Pr[(A\cap B)^c]$$