Question
A standard deck of cards has 52 cards. You choose one card from the deck. Let event A be you choose a black card and event B be you choose a king. Find P(A)⋅P(B) , P(A and B) , and determine whether events A and B are independent. Enter the first two answers in fraction form. (2 points) P(A)⋅P(B)= ; P(A and B)= ; Enter 1 if events A and B are dependent. Enter 2 if events A and B are independent
Answers
GPT-4o mini
To solve this problem, we first need to calculate the probabilities for the events A and B.
1. **Event A**: Choosing a black card.
- There are 26 black cards in a standard 52-card deck (13 spades and 13 clubs).
- Therefore, \( P(A) = \frac{26}{52} = \frac{1}{2} \).
2. **Event B**: Choosing a king.
- There are 4 kings in a standard 52-card deck (one from each suit).
- Therefore, \( P(B) = \frac{4}{52} = \frac{1}{13} \).
3. **Calculating \( P(A) \cdot P(B) \)**:
\[
P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{13} = \frac{1}{26}.
\]
4. **Finding \( P(A \text{ and } B) \)**:
- The event \( A \text{ and } B \) corresponds to choosing a black king. There are 2 black kings (the king of spades and the king of clubs).
- Therefore, \( P(A \text{ and } B) = \frac{2}{52} = \frac{1}{26} \).
5. **Determining if A and B are independent**:
- Two events A and B are independent if \( P(A \text{ and } B) = P(A) \cdot P(B) \).
- Here, \( P(A \text{ and } B) = \frac{1}{26} \) and \( P(A) \cdot P(B) = \frac{1}{26} \).
- Since these are equal, 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 (enter 2).
Final answers:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \text{ and } B) = \frac{1}{26} \)
- Independence of events: 2
1. **Event A**: Choosing a black card.
- There are 26 black cards in a standard 52-card deck (13 spades and 13 clubs).
- Therefore, \( P(A) = \frac{26}{52} = \frac{1}{2} \).
2. **Event B**: Choosing a king.
- There are 4 kings in a standard 52-card deck (one from each suit).
- Therefore, \( P(B) = \frac{4}{52} = \frac{1}{13} \).
3. **Calculating \( P(A) \cdot P(B) \)**:
\[
P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{13} = \frac{1}{26}.
\]
4. **Finding \( P(A \text{ and } B) \)**:
- The event \( A \text{ and } B \) corresponds to choosing a black king. There are 2 black kings (the king of spades and the king of clubs).
- Therefore, \( P(A \text{ and } B) = \frac{2}{52} = \frac{1}{26} \).
5. **Determining if A and B are independent**:
- Two events A and B are independent if \( P(A \text{ and } B) = P(A) \cdot P(B) \).
- Here, \( P(A \text{ and } B) = \frac{1}{26} \) and \( P(A) \cdot P(B) = \frac{1}{26} \).
- Since these are equal, 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 (enter 2).
Final answers:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \text{ and } B) = \frac{1}{26} \)
- Independence of events: 2