= Solution
\b[The variables are not independent.] Their joint support excludes $X>1/2$ and $Y>1/2$ simultaneously, since every point with positive joint density satisfies $X+Y\leq1$. Yet each event separately has positive <probability>: the density is strictly positive on open triangular subregions where $X>1/2$, and similarly where $Y>1/2$. Thus
$$
\Pr(X>1/2,Y>1/2)=0<\Pr(X>1/2)\Pr(Y>1/2).
$$
This contradicts the event factorization required for <independence of random variables>. The product-shaped formula $24xy$ on its support does not undo the dependence created by the triangular support.
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