Conditional Probability, Independence & Bayes · Challenge problem

A uniformly random bit string $b_1b_2b_3\in\{0,1\}^3$ is generated. Let $A=\{b_1=b_2\}$, $B=\{b_2=b_3\}$, $C=\{b_1=b_3\}$. Find $P(A\cap B\cap C)$ and decide whether $A,B,C$ are mutually independent (they are pairwise independent).