Divisibility Events and the Containment Trap

Let $\Omega = \{0, 1, 2, \ldots, 11\}$ with uniform probability $P(\{k\}) = 1/12$ for each $k$. Define three events based on divisibility: $A = \{k \in \Omega : 2 \mid k\}$ (even numbers), $B = \{k \in \Omega : 3 \mid k\}$ (multiples of $3$), $C = \{k \in \Omega : 4 \mid k\}$ (multiples of $4$). (a) List each event explicitly and compute $P(A)$, $P(B)$, $P(C)$. (b) For each of the three pairs $(A,B)$, $(A,C)$, $(B,C)$, determine whether the pair is independent by computing $P(\text{intersection})$ and comparing with the product of marginals. (c) Identify which pair fails independence and explain the structural reason.