We have 4 cards on a table: $\displaystyle \spadesuit K$, $\displaystyle \spadesuit Q$, $\displaystyle \heartsuit K$ and $\displaystyle \heartsuit Q$.

We blindly choose one card.

Define events as:

$\displaystyle A \equiv \spadesuit$ chosen

$\displaystyle B \equiv Q$ chosen

$\displaystyle C \equiv \heartsuit K \cup \spadesuit Q$ chosen

Are following events independent?

- $\displaystyle A \cap B$
- $\displaystyle A \cap C$
- $\displaystyle B \cap C$
- $\displaystyle A \cap B \cap C$

- Yes.
- Yes: $\displaystyle P(A \cap C)=P(A)*P(C|A)=\frac{1}{2}\frac{1}{2}==P(A)*P(C)=\ frac{1}{2}\frac{1}{2}$
- Yes: $\displaystyle P(B \cap C)=P(B)*P(C|B)=\frac{1}{2}\frac{1}{2}==P(B)*P(C)=\ frac{1}{2}\frac{1}{2}$
[Solutions say "No."]- No: $\displaystyle P(A \cap B \cap C)=P(A)*P(B|A)*P(C|A\cap B)=\frac{1}{2}\frac{1}{2}1\neq P(A)*P(B)*P(C)=\frac{1}{2}\frac{1}{2}\frac{1}{2}$

Please, also tell me whether points 2. and 4. are correct (for the right reasons)?

Thank you!