I got stuck trying to figuring out how to show the following question in probabilistic theory:
"We say that A and B (with P(A), P(B) > 0) attract each other when P(A|B) > P(A)." I've shown that when A attracts B, B also attracts A.
But now I have problems showing that when B is attracted to A with P(NOT B) > 0, NOT B is not attracted to A. ( P(NOT B | A) > P(NOT B) should not be true )
I've tried to make use of the formula P(NOT B | A) = 1 - P(B|A) but there was no real success. Maybe someone has an idea about the technique of proof? My idea was to assume that P(NOT B | A) > P(NOT B) is true and then show that its not true because of something.
Thanks for your help!
You assume : $P(B|A)>P(B)$
so: $-P(B|A)<-P(B)$
so: $1-P(B|A)<1-P(B)$
The complement rule for conditional probabilities tells us : $1-P(B|A)=P(NOT B|A)$
so: $P(NOT B|A)<P(NOTB)$
so NOT B is not attracted to A