Expectation of $C|D$ if we know $P(S)$ and $P(C|S)$ and $P(D|S)$

Question

Given if we know $P(S)$ and $P(C|S)$ and $P(D|S)$, how do you compute $E[C|D=d]$? One way that I thought of is to find the conditional probability of $P(C|D)$ by computing the joint probability $P(C,D,S)$ and marginalizing it over $S$. But, $P(D|S)$ is a binomial distribution with parameter $q$ and $S$. Finding the full joint probability distribution will be too complicated. Does anyone know an easier way to find $E[C|D=d]$? Thanks.

Note: I forgot to mention that C and D are independent given S

Answer 1

You have $$ P(C\cap S)=P(S)P(C|S), P(D\cap S)=P(S)P(D|S) $$ This information is not enough to let you compute $$ P(C\cap D), P(C|D) $$ because you lost information at $$ P(C\cap D\cap S^{c}) $$ and in principle you do not know what this is.

Answer 2

$$ E(C\mid D=d)=\frac{E(C;D=d)}{P(D=d)} $$ $$ E(C;D=d)=\sum_{c,s}c\,P(C=c\mid S=s)\,P(D=d\mid S=s)\,P(S=s) $$ $$ P(D=d)=\sum_{s}P(D=d\mid S=s)\,P(S=s) $$