In a Bayesian network, is $P(J|B)$ the same as $P(B|J)$?

Question

In a network like this:

Is $P(J|B)$ the same as $P(B|J)$?

Since $P(J|B)=\sum _{a,e}P(B,J,A=a,E=e)= \alpha P(B)\sum _{a,e}P(J|A)P(A|B,E)P(E) = P(B|J)$

Because $P(B|J)=\sum _{a,e}P(B,J,A=a,E=e)$?

Answer 1

Not necessarily, because $$P(J|B)=\frac{P(J,B)}{P(B)}=\color{red}{\frac{1}{P(B)}}\sum\limits_{a,e} P(B,J,A=a,E=e)$$ and $$P(B|J)=\frac{P(J,B)}{P(J)}=\color{red}{\frac{1}{P(J)}}\sum\limits_{a,e} P(B,J,A=a,E=e)$$