Pairwise and Triple Independence Without Mutual Independence of Four Events

Let $\Omega$ consist of all binary strings of length $4$ with an even number of $1$s, each equally likely: $$\Omega = \{0000,\, 0011,\, 0101,\, 0110,\, 1001,\, 1010,\, 1100,\, 1111\}.$$ Define events $A_i = \{\omega \in \Omega : \omega_i = 1\}$ for $i = 1,2,3,4$.

(a) Show that $P(A_i) = 1/2$ for each $i$.

(b) Verify that every pair is independent: $P(A_i \cap A_j) = 1/4$ for all $i \neq j$.

(c) Verify that every triple is independent: $P(A_i \cap A_j \cap A_k) = 1/8$ for all distinct $i,j,k$.

(d) Compute $P(A_1 \cap A_2 \cap A_3 \cap A_4)$ and compare with $P(A_1) P(A_2) P(A_3) P(A_4)$. Are the four events mutually independent?