We have three events A, B, and C such that
P(A|C)\geqP(B|C) and P(A|C')\geqP(B|C')
Prove that P(A)\geqP(B).
First I started with,
P(A|C)=P(AC)/P(C) and similarly P(B|C)=P(BC)/P(C).
From above,
P(AC)/P(C)\geqP(BC)/P(C) and since P(C)\geq0,
P(AC)\geqP(BC).
Similarly from above...