A hand with 5 cards is dealt from a deck of 52 cards.
Let A be the event that the hand is a four of a kind (4 cards of one rank and a 5th card)
Let B be the event that at least one of the cards in the hand is a queen
Are A and B independent? If so, why?
Let $A$ be the event the hand is a four-of-a-kind. Let $B$ be the event the hand contains at least one Queen.
Approach via direct counting principles.
The probability the hand is a four-of-a-kind: Pick the rank of the quadruple. Pick the remaining card.
$$Pr(A)=\dfrac{13\times 48}{\binom{52}{5}}$$
The probability the hand contains at least one Queen: This is the opposite probability than the probability that the hand contains no Queens. Choose the five non-queen cards.
$$Pr(B)=1-Pr(B^c) = 1 - \dfrac{\binom{48}{5}}{\binom{52}{5}}$$
The probability the hand contains at least one Queen and is a four-of-a-kind: Pick whether the queen was the four-of-a-kind or the kicker. Pick the other rank appearing. Pick the suit of the kicker.
$$Pr(A\cap B) = \dfrac{2\times 12\times 4}{\binom{52}{5}}$$
You can verify now that $Pr(A\cap B)\neq Pr(A)\times Pr(B)$ thus showing they are dependent.