To solve this problem, we first need to find the probabilities of events A and B, and then check for independence.
- Determine \( P(A \text{ and } B) \):
- Event A is choosing a red card. There are 26 red cards (13 diamonds + 13 hearts) in the deck.
- Event B is choosing an ace. There are 4 aces total (one in each suit).
- The intersection \( A \text{ and } B \) (choosing a card that is both red and an ace) includes the two red aces: the Ace of Diamonds and the Ace of Hearts. So, there are 2 outcomes that are both in A and B.
\[ P(A \text{ and } B) = \frac{\text{Number of favorable outcomes for } A \text{ and } B}{\text{Total number of outcomes}} = \frac{2}{52} = \frac{1}{26} \]
- Determine \( P(A) \):
- As mentioned, there are 26 red cards.
\[ P(A) = \frac{\text{Number of favorable outcomes for } A}{\text{Total number of outcomes}} = \frac{26}{52} = \frac{1}{2} \]
- Determine \( P(B) \):
- There are 4 aces in total.
\[ P(B) = \frac{\text{Number of favorable outcomes for } B}{\text{Total number of outcomes}} = \frac{4}{52} = \frac{1}{13} \]
- Check for Independence:
- Two events A and B are independent if \( P(A \text{ and } B) = P(A) \times P(B) \).
Calculating \( P(A) \times P(B) \):
\[ P(A) \times P(B) = \frac{1}{2} \times \frac{1}{13} = \frac{1}{26} \]
Since \( P(A \text{ and } B) = P(A) \times P(B) \), events A and B are independent.
Therefore, the answers are:
- \( P(A \text{ and } B) = \frac{1}{26} \)
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{13} \)
- Events A and B are independent, so enter 2.
Final answers:
- \( P(A \text{ and } B) = \frac{1}{26} \)
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{13} \)
- Events A and B are independent: 2.