I there, about this exercise:
An event F is said to carry negative information about an event E, and we write $F\rightarrow E$ if $P(E|F)\leq P(E)$
Prove or give a counter example for the following claim:
If $F\rightarrow E$ then $E \rightarrow F $
Well, my attempt was:
$$ P(E|F) = \frac{P(E \cap F)}{P(F)} \leq P(E) => P(E \cap F)\geq P(E)*P(F) $$
$$ P(F|E) = \frac{P(F \cap E)}{P(E)} \leq P(F) => P(F \cap E)\geq P(F)*P(E) $$
And since $ P(F \cap E) = P(E \cap F) $ Than it seems pretty correct to me! (but the solution manual says I'm wrong).
What is wrong in my way of thinking? Thanks!
The $\geq$ signs in your attempt should be $\leq$ signs.
That's all I can find.
Further the idea is okay.
If both sides are multiplied by factor $P(F)$ then $P(E\mid F)\leq P(E)$ changes into: $$P(E\cap F)\leq P(E)P(F)$$and it is immediate then that the relation is symmetric.