Q.A card is drawn from a pack of $52$. Are the events $A$ = "the card is a king" and $B$ = "the card is a heart" independent? Justify using probabilities.

Test independence by checking whether $P(A\cap B)=P(A)\,P(B)$.

$P(A)P(B)=\dfrac{4}{52}\cdot\dfrac{13}{52}=\dfrac{1}{52}=P(A\cap B)$, so they are independent.

There are $4$ kings, $13$ hearts, and exactly one card that is both (the king of hearts). So

Independence test. Compute the product:

Since $P(A\cap B)=\dfrac{1}{52}=P(A)\,P(B)$, the events satisfy the independence condition and are therefore independent.

Independent check via conditional probability. $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{1/52}{1/4}=\dfrac{1}{13}=P(A)$. Knowing the card is a heart does not change the probability it is a king, confirming independence.

Alternatively check $P(A\mid B)=P(A)$: $P(A\mid B)=1/13=P(A)$, the same conclusion.

Assuming that because a card cannot be examined "twice" the events must be dependent, without doing the test. Independence is a numerical condition ($P(A\cap B)=P(A)P(B)$), not a guess — and here it happens to hold exactly.