To solve for the probabilities \( P(A) \) and \( P(B) \), and determine the independence of the events, we start with identifying the events and their outcomes in a standard deck of 52 cards.
Event A: You choose a black card.
There are 26 black cards in a standard deck (13 spades + 13 clubs).
Thus, the probability of selecting a black card is:
\[
P(A) = \frac{26}{52} = \frac{1}{2}
\]
Event B: You choose a king.
There are 4 kings in the deck (one in each suit).
Thus, the probability of selecting a king is:
\[
P(B) = \frac{4}{52} = \frac{1}{13}
\]
Now we can calculate \( P(A) \cdot P(B) \): \[ P(A) \cdot P(B) = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{13}\right) = \frac{1}{26} \]
Next, we find \( P(A \text{ and } B) \), which represents the probability of selecting a card that is both black and a king.
There are 2 black kings (King of Spades and King of Clubs) in the deck.
Thus,
\[
P(A \text{ and } B) = \frac{2}{52} = \frac{1}{26}
\]
Now we check if events A and B are independent. Two events A and B are independent if: \[ P(A \text{ and } B) = P(A) \cdot P(B) \]
We already have:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \text{ and } B) = \frac{1}{26} \)
Since \( P(A \text{ and } B) = P(A) \cdot P(B) \), the events A and B are independent.
Putting it all together, we have:
\[
P(A) \cdot P(B) = \frac{1}{26}
\]
\[
P(A \text{ and } B) = \frac{1}{26}
\]
Events A and B are independent, so we enter 2 as our final answer.
Thus, the final answers are:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \text{ and } B) = \frac{1}{26} \)
- Events A and B are independent:
2