Identical Marginals Do Not Imply Independence

Three cards are labeled $(0,0)$, $(0,1)$, and $(1,0)$. One card is drawn uniformly at random. Let $X$ be the first number and $Y$ the second number on the drawn card. (a) Compute $P(X = 0)$, $P(X = 1)$, $P(Y = 0)$, and $P(Y = 1)$. (b) Are the marginal distributions of $X$ and $Y$ the same? (c) Are $X$ and $Y$ independent? Verify by checking whether $P(X = x, Y = y) = P(X = x) P(Y = y)$ holds for all $(x, y) \in \{0,1\}^2$.