Question
A standard deck of cards has 52 cards. The 52 cards are divided into 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
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 solve this problem, we first need to find the probabilities of events A and B, and then check for independence.
1. **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.
\[
P(A \text{ and } B) = \frac{\text{Number of favorable outcomes for } A \text{ and } B}{\text{Total number of outcomes}} = \frac{2}{52} = \frac{1}{26}
\]
2. **Determine \( P(A) \)**:
- As mentioned, there are 26 red cards.
\[
P(A) = \frac{\text{Number of favorable outcomes for } A}{\text{Total number of outcomes}} = \frac{26}{52} = \frac{1}{2}
\]
3. **Determine \( P(B) \)**:
- There are 4 aces in total.
\[
P(B) = \frac{\text{Number of favorable outcomes for } B}{\text{Total number of outcomes}} = \frac{4}{52} = \frac{1}{13}
\]
4. **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**.
1. **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.
\[
P(A \text{ and } B) = \frac{\text{Number of favorable outcomes for } A \text{ and } B}{\text{Total number of outcomes}} = \frac{2}{52} = \frac{1}{26}
\]
2. **Determine \( P(A) \)**:
- As mentioned, there are 26 red cards.
\[
P(A) = \frac{\text{Number of favorable outcomes for } A}{\text{Total number of outcomes}} = \frac{26}{52} = \frac{1}{2}
\]
3. **Determine \( P(B) \)**:
- There are 4 aces in total.
\[
P(B) = \frac{\text{Number of favorable outcomes for } B}{\text{Total number of outcomes}} = \frac{4}{52} = \frac{1}{13}
\]
4. **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**.