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

Answer 1

To determine \( P(A \text{ and } B) \), \( P(A) \), \( P(B) \), and the independence of events \( A \) and \( B \), we first need to define the events clearly:

- **Event A:** Choosing a red card. There are 26 red cards in a standard deck (13 diamonds and 13 hearts).
- **Event B:** Choosing an ace. There are 4 aces in a standard deck (one for each suit).

### Step 1: Find \( P(A) \)
The probability of choosing a red card is:
\[
P(A) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}.
\]

### Step 2: Find \( P(B) \)
The probability of choosing an ace is:
\[
P(B) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}.
\]

### Step 3: Find \( P(A \text{ and } B) \)
Event \( A \text{ and } B \) refers to the probability of choosing a card that is both red and an ace. There are 2 red aces in a deck (the Ace of Diamonds and the Ace of Hearts). So, the probability is:
\[
P(A \text{ and } B) = \frac{\text{Number of red aces}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}.
\]

### Step 4: Check Independence
To check whether events \( A \) and \( B \) are independent, we need to verify if:
\[
P(A \text{ and } B) = P(A) \cdot P(B).
\]
Calculating \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{13}\right) = \frac{1}{26}.
\]
Since \( P(A \text{ and } B) = \frac{1}{26} \) and \( P(A) \cdot P(B) = \frac{1}{26} \), the events are independent.

### Final Results
\[
P(A \text{ and } B) = \frac{1}{26}, \quad P(A) = \frac{1}{2}, \quad P(B) = \frac{1}{13}
\]
Enter 1 for dependent or 2 for independent: 2.

Thus, the final 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 (2).