= Solution
\b[The two outcomes are not <independent>.] Conditional on the selected die, the rolls are <independent>, but sharing the unknown die creates <statistical dependence>. Each nonfavored face has <probability> $(1-1/2)/5=1/10$.
The unconditional <marginal distribution> assigns the following <probability> to a $1$ on either roll:
$$
\mathbb P(X=1)=\mathbb P(Y=1)=
\frac16\left(\frac12+5\cdot\frac1{10}\right)=\frac16.
$$
However, by the <law of total probability>,
$$
\mathbb P(X=1,Y=1)=
\frac16\left[\left(\frac12\right)^2+
5\left(\frac1{10}\right)^2\right]=\frac1{20}\ne\frac1{36}.
$$
Thus the joint <probability> does not factor into the two marginals. <Conditional independence> given a shared latent choice does not imply unconditional <independence>.
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