# Is this Bayesian Network Probability calculation correct?

I think I understand how to calculate BN and why it is so, but complex net still confuses me. Currently how I understand it is that, if there is any 'result' variable in the probability, it can be calculated by converting it to P(result|cause)P(cause). So I assumed in this bayesian network (A,B → C → D,E), if you want to get P(B,E), every parent and grand parent of B and E should be considered. As B doesn't have parent, only the (grand) parent of E (A,C) will hop in, and finally it can be calculated as :

P(B,E) = P(B,E,A,C)+P(B,E,~A,C)+P(B,E,A,~C)+P(B,E,~A,~C)

The joint probability would be :

P(B,E,A,C) = P(B)P(A)P(C|A,B)P(E|C)

The rest of them are calculated the same way. I'd like to check if this calculation and my understanding is right. Thank you!

For the DAG: $$\begin{array}{ccc}A&&&&B\\&\searrow&&\swarrow \\&&C\\&\swarrow&&\searrow\\D&&&&E\end{array}$$
Then, yes,$$P_{B,E}=\sum_{A,C}P_AP_BP_{C\mid A, B}P_{E\mid C}$$