P and S are the common cause of c. If P(C=true| P,S ) is given , can I introduce S to P(C|P) as P(C= true|P= true)= P(C=true| P =true , S= true)* P(P=true ,S=true )+ P(C= true | P=true ,S =false )*P(P=true ,S =false) ? Is this a correct way of using the sum rule? Many thanks.

  • $\begingroup$ The notation is confusing. Can you use some symbol other than P for denoting one of the common causes of c? Because you are also using P to denote probability and what you are writing is confusing because of the overloading of P. -1 pending editing the question to use more rational nomenclature. The use of LaTEx/MathJax would also help. @gunes in the hope that he will amend his answer. $\endgroup$ – Dilip Sarwate May 26 '19 at 16:34

That's not correct. The correct way to do it is (letting $\text{t}$ be true, and $\text{f}$ be false): $$\begin{align}P(C=\text{t}|P=\text{t})=\sum_{S_i\in \{\text{t},\ \text{f}\}}P(C=\text{t}|P=\text{t},S=S_i)P(S=S_i|P=\text{t})\end{align}$$

The intuition is thinking that $P$ is always given. Never transfer it to the left of $\vert$ sign. For example, if you remove $P$ out of the equation above, you'll directly obtain the Total Probability Law.

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