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 be you choose a
red card and event you
choose an ace. Determine
, , and
, and state whether
events and are
independent.
Enter the first three answers
in fraction form. Reduce
your answer.
(2 points)
A
B
P (A and B) P (A)
P (B)
A B
P (A and B) =
P (A) =
P (B) =
Enter 1 if events A and B are dependent. Enter 2 if events A and B aree independent.

Answer 1

Let's define the events clearly and calculate the probabilities involved.

- **Event A**: Choosing a red card.
- **Event B**: Choosing an ace.

### Step 1: Calculate \( P(A) \)

There are 26 red cards in total (13 diamonds and 13 hearts). Thus:

\[
P(A) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}
\]

### Step 2: Calculate \( P(B) \)

There are 4 aces in total (one from each suit). Thus:

\[
P(B) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}
\]

### Step 3: Calculate \( P(A \text{ and } B) \)

The only red aces are the Ace of Hearts and the Ace of Diamonds, so there are 2 red aces. Thus:

\[
P(A \text{ and } B) = \frac{\text{Number of red aces}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}
\]

### Summary of Probabilities

- \( P(A \text{ and } B) = \frac{1}{26} \)
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{13} \)

### Step 4: Check Independence

Events A and B are independent if:

\[
P(A \text{ and } B) = P(A) \cdot P(B)
\]

Calculate \( P(A) \cdot P(B) \):

\[
P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{13} = \frac{1}{26}
\]

Since \( P(A \text{ and } B) = \frac{1}{26} \) and \( P(A) \cdot P(B) = \frac{1}{26} \), the events A and B are independent.

### 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

Here are the final entries in the required format:

\[
P(A \text{ and } B) = \frac{1}{26}, \quad P(A) = \frac{1}{2}, \quad P(B) = \frac{1}{13}, \quad \text{(Independent)} = 2
\]