To determine the probabilities and the independence of events A and B, we start by defining the events clearly:
- Event A: Choosing a red card. There are 26 red cards in a deck (13 diamonds + 13 hearts).
- Event B: Choosing an ace. There are 4 aces in a deck (1 in each suit).
Next, we find the intersections and individual probabilities.
1. Calculate P(A and B): Event A and B (P(A and B)): This is the probability of choosing a card that is both a red card and an ace. There are 2 red aces (Ace of Hearts and Ace of Diamonds) in the deck.
\[ P(A \text{ and } B) = \frac{\text{Number of red aces}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26} \]
2. Calculate P(A): P(A) is the probability of choosing a red card. There are 26 red cards out of 52 total cards.
\[ P(A) = \frac{26}{52} = \frac{1}{2} \]
3. Calculate P(B): P(B) is the probability of choosing an ace. There are 4 aces out of 52 total cards.
\[ P(B) = \frac{4}{52} = \frac{1}{13} \]
4. Check for independence: Two events A and B are independent if:
\[ P(A \text{ and } B) = P(A) \cdot P(B) \]
We calculate \(P(A) \cdot P(B)\):
\[ P(A) \cdot P(B) = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{13}\right) = \frac{1}{26} \]
Since \(P(A \text{ and } B) = P(A) \cdot P(B)\), the events A and B are independent.
Now, summarizing the results:
\[ P(A \text{ and } B) = \frac{1}{26} \] \[ P(A) = \frac{1}{2} \] \[ P(B) = \frac{1}{13} \] \[ \text{Enter 1 if dependent, 2 if independent:} \quad 2 \]
Final answers:
- \( P(A \text{ and } B) = \frac{1}{26} \)
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{13} \)
- Independence: \( 2 \)
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