Fixed Points of a Random Permutation Are Not Independent

A permutation $\sigma$ of $\{1,2,3,4\}$ is chosen uniformly at random (each of the $4! = 24$ permutations equally likely). For $i = 1,2,3,4$, define the event $A_i = \{\sigma(i) = i\}$ (element $i$ is a fixed point). (a) By counting, show that $P(A_i) = 1/4$ for every $i$, and $P(A_i \cap A_j) = 1/12$ for every pair $i \neq j$. (b) Are $A_i$ and $A_j$ independent? (c) Compute $P(A_i \cap A_j \cap A_k)$ for distinct $i,j,k$ and $P(A_1 \cap A_2 \cap A_3 \cap A_4)$. (d) Despite the failure of pairwise independence, verify the classical inclusion-exclusion identity: $P\bigl(\bigcup_{i=1}^{4} A_i\bigr) = 1 - \frac{1}{2!} + \frac{1}{3!} - \frac{1}{4!}$.