Question

P(A) = 1/4, P(B) = 2/3, P(A and B) = 1/6. Are Events A and B independent?

Answer 1

For $A$ and $B$ to be independent, it must be the case that

$\mathbb P(A\cap B)=\mathbb P(A)\times\mathbb P(B)$

You're given

$\mathbb P(A\cap B)=\dfrac16$

while

$\mathbb P(A)\times\mathbb P(B)=\dfrac14\times\dfrac23=\dfrac16$

So yes, the two events are independent.