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∩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.
(8 points)
, 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.
(8 points)
Answers
GPT-4o mini
To determine \( P(A \cap B) \), \( P(A) \), and \( P(B) \), we first need to define the two events clearly:
- Event \( A \): Choosing a red card (diamonds or hearts).
- Event \( B \): Choosing an ace.
### Step 1: Calculate \( P(A) \)
The total number of red cards in a standard deck is 26 (13 diamonds and 13 hearts).
Hence, the probability \( P(A) \) is:
\[
P(A) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}
\]
### Step 2: Calculate \( P(B) \)
The total number of aces in the deck is 4 (one from each suit).
Thus, the probability \( P(B) \) is:
\[
P(B) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}
\]
### Step 3: Calculate \( P(A \cap B) \)
Event \( A \cap B \) refers to choosing a card that is both red and an ace. There are 2 red aces in the deck (the Ace of Diamonds and the Ace of Hearts).
So, the probability \( P(A \cap B) \) is:
\[
P(A \cap B) = \frac{\text{Number of red aces}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}
\]
### Step 4: Check for Independence
Two events \( A \) and \( B \) are independent if:
\[
P(A \cap B) = P(A) \times P(B)
\]
Now we calculate \( P(A) \times P(B) \):
\[
P(A) \times P(B) = \left(\frac{1}{2}\right) \times \left(\frac{1}{13}\right) = \frac{1}{26}
\]
Since \( P(A \cap B) = P(A) \times P(B) \), the events \( A \) and \( B \) are independent.
### Summary of Results
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{13} \)
- \( P(A \cap B) = \frac{1}{26} \)
### Final Answer
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{13} \)
- \( P(A \cap B) = \frac{1}{26} \)
Events A and B are independent.
- Event \( A \): Choosing a red card (diamonds or hearts).
- Event \( B \): Choosing an ace.
### Step 1: Calculate \( P(A) \)
The total number of red cards in a standard deck is 26 (13 diamonds and 13 hearts).
Hence, the probability \( P(A) \) is:
\[
P(A) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}
\]
### Step 2: Calculate \( P(B) \)
The total number of aces in the deck is 4 (one from each suit).
Thus, the probability \( P(B) \) is:
\[
P(B) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}
\]
### Step 3: Calculate \( P(A \cap B) \)
Event \( A \cap B \) refers to choosing a card that is both red and an ace. There are 2 red aces in the deck (the Ace of Diamonds and the Ace of Hearts).
So, the probability \( P(A \cap B) \) is:
\[
P(A \cap B) = \frac{\text{Number of red aces}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}
\]
### Step 4: Check for Independence
Two events \( A \) and \( B \) are independent if:
\[
P(A \cap B) = P(A) \times P(B)
\]
Now we calculate \( P(A) \times P(B) \):
\[
P(A) \times P(B) = \left(\frac{1}{2}\right) \times \left(\frac{1}{13}\right) = \frac{1}{26}
\]
Since \( P(A \cap B) = P(A) \times P(B) \), the events \( A \) and \( B \) are independent.
### Summary of Results
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{13} \)
- \( P(A \cap B) = \frac{1}{26} \)
### Final Answer
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{13} \)
- \( P(A \cap B) = \frac{1}{26} \)
Events A and B are independent.