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}).

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