= Solution
Put $S=X+Y$. Given $X$, its conditional <expected value> is $3X/2$ and its conditional <variance> is $X/4$, since $X$ is then fixed. The <law of total expectation> and <law of total variance> give
$$
\boxed{\mathbb E S=\frac{3(1-p)}{2p},\qquad
\operatorname{Var}(S)=\frac14\mathbb E X+\frac94\operatorname{Var}(X)
=\frac{(1-p)(p+9)}{4p^2}.}
$$
The two scores are dependent: the <law of total covariance> gives $\operatorname{Cov}(X,Y)=\operatorname{Cov}(X,X/2)=q/(2p^2)$. Adding their two <variances> without that <covariance> term would give an incorrect total.
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