Three cards labeled $A$, $B$, and $C$ are shuffled and laid out in a row from left to right. Let $E_1$ be the event "Card $A$ is placed to the left of Card $B$" and $E_2$ be the event "Card $B$ is placed to the left of Card $C$". Evaluate $P(E_1 \cap E_2)$ and determine whether $E_1$ and $E_2$ are independent.