Independence Is Fragile Under Sample-Space Perturbation

A four-card deck contains $A\spadesuit$, $A\heartsuit$, $K\spadesuit$, $K\heartsuit$. One card is drawn uniformly at random. Let $R$ = 「the card is an Ace」 and $S$ = 「the card is a Spade.」 (a) Compute $P(R)$, $P(S)$, $P(R \cap S)$ and verify that $R$ and $S$ are independent. (b) Now remove $K\heartsuit$ from the deck, leaving a three-card deck $\{A\spadesuit, A\heartsuit, K\spadesuit\}$ with uniform draw. Recompute $P(R)$, $P(S)$, $P(R \cap S)$ and determine whether $R$ and $S$ are still independent. (c) Explain intuitively why removing a single card destroyed independence.