Bounding probability of an event

1.

– If  $A \Rightarrow B$ then $Pr(A) \leq Pr(B)$.

– If $A \Rightarrow B$ and $A \Rightarrow C$ then $Pr(A) \leq \frac{1}{2}(Pr(B) +Pr(C))$.

(Note that neither $B$ and $C$ necessarily imply $A$ nor $B$ and $C$ are mutually exclusive.)

More generally, if we can prove that event $A$ implies one of $n$ events $B_{1},B_{2}, \ldots, B_{n}$ (it is not necessary that $A$ can only be one of $B_{1},B_{2}, \ldots , B_{n}$ at one time.)then $Pr(A) \leq \frac{1}{n}\sum_{i=1}^{n}Pr(B_{i})$.