let AB and C be three events in the sample space S and assum

let A,B and C be three events in the sample space S and assume the following is true:

P(A|C)>=P(B|C) and P(A|C\')>=(B|C\').

Prove (A)>=P(B)

Solution

In the complete sample space S an event may can either have occured or not occured within sample space C. ie it can either be C or C\'.

It is given that P(A|C)>=P(B|C) as well as P(A|C\')>=P(B|C\'),

hence irrespective of occurence of C, P(A)>=P(B)