Answer
Not independent
Work Step by Step
We know that both $A$ and $B$ are independent events when it satisfies the following condition such that $P(A \cap B)=P(A) P(B) ~~~(1)$
Here, $A$ denotes both dice are $1$ and $B$ denotes neither die is $2$.
We can see that $P(A)=\dfrac{n(A)}{n(S)}=\dfrac{1}{36}$ and
$P(B)=\dfrac{n(B)}{n(S)}=\dfrac{25}{36}$
Next, we find that there are only one outcome that can be possible.
So, $P(A \cap B)=\dfrac{n(A \cap B)}{n(S)}=\dfrac{1}{36}$
Also, $P(A \cap B)=\dfrac{1}{36}$ and $P(A) P(B)=(\dfrac{1}{36})(\dfrac{5}{18})=\dfrac{25}{1296}$
This means that condition (1) is satisfied. So, both $A$ and $B$ are Not independent events.