Question
four suits: diamonds, clubs, hearts, and spades. Diamonds and hearts are red cards, and clubs and spades are black cards. Suppose you choose a card from the deck. Let event A
be you choose a red card and event B
you choose an ace. Determine P(A and B)
, P(A)
, and P(B)
, and state whether events A
and B
are independent.
Enter the first three answers in fraction form. Reduce your answer.
(2 points)
P(A and B)=
;
P(A)=
;
P(B)=
;
Enter 1 if events A
and B
are dependent. Enter 2 if events A
and B
are independent
be you choose a red card and event B
you choose an ace. Determine P(A and B)
, P(A)
, and P(B)
, and state whether events A
and B
are independent.
Enter the first three answers in fraction form. Reduce your answer.
(2 points)
P(A and B)=
;
P(A)=
;
P(B)=
;
Enter 1 if events A
and B
are dependent. Enter 2 if events A
and B
are independent
Answers
GPT-4o mini
To answer your question, let's analyze the problem step-by-step using a standard 52-card deck.
1. **Identify the events:**
- Event A: Choosing a red card (diamonds or hearts).
- Event B: Choosing an ace.
2. **Determine the probabilities:**
- There are 26 red cards (13 diamonds + 13 hearts) in a standard deck of 52 cards.
- There are 4 aces in total (1 ace of diamonds, 1 ace of hearts, 1 ace of clubs, 1 ace of spades).
3. **Calculate \( P(A \cap B) \) (the probability of choosing a card that is both red and an ace):**
- The only red aces are the ace of diamonds and the ace of hearts.
- So, there are 2 cards that are both red and aces.
\[
P(A \cap B) = \frac{\text{Number of red aces}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}
\]
4. **Calculate \( P(A) \) (the probability of choosing a red card):**
\[
P(A) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}
\]
5. **Calculate \( P(B) \) (the probability of choosing an ace):**
\[
P(B) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}
\]
6. **Determine if events A and B are independent:**
- Two events A and B are independent if \( P(A \cap B) = P(A) \times P(B) \).
- Calculate \( P(A) \times P(B) \):
\[
P(A) \times P(B) = \frac{1}{2} \times \frac{1}{13} = \frac{1}{26}
\]
- Since \( P(A \cap B) = \frac{1}{26} \) matches \( P(A) \times P(B) = \frac{1}{26} \), the events A and B are independent.
**Final answers:**
- \( P(A \cap B) = \frac{1}{26} \)
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{13} \)
- Since A and B are independent, we enter **2**.
1. **Identify the events:**
- Event A: Choosing a red card (diamonds or hearts).
- Event B: Choosing an ace.
2. **Determine the probabilities:**
- There are 26 red cards (13 diamonds + 13 hearts) in a standard deck of 52 cards.
- There are 4 aces in total (1 ace of diamonds, 1 ace of hearts, 1 ace of clubs, 1 ace of spades).
3. **Calculate \( P(A \cap B) \) (the probability of choosing a card that is both red and an ace):**
- The only red aces are the ace of diamonds and the ace of hearts.
- So, there are 2 cards that are both red and aces.
\[
P(A \cap B) = \frac{\text{Number of red aces}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}
\]
4. **Calculate \( P(A) \) (the probability of choosing a red card):**
\[
P(A) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}
\]
5. **Calculate \( P(B) \) (the probability of choosing an ace):**
\[
P(B) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}
\]
6. **Determine if events A and B are independent:**
- Two events A and B are independent if \( P(A \cap B) = P(A) \times P(B) \).
- Calculate \( P(A) \times P(B) \):
\[
P(A) \times P(B) = \frac{1}{2} \times \frac{1}{13} = \frac{1}{26}
\]
- Since \( P(A \cap B) = \frac{1}{26} \) matches \( P(A) \times P(B) = \frac{1}{26} \), the events A and B are independent.
**Final answers:**
- \( P(A \cap B) = \frac{1}{26} \)
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{13} \)
- Since A and B are independent, we enter **2**.