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.

Answer 1

To find \( P(A \text{ and } B) \), \( P(A) \), and \( P(B) \), we first define the events:

- Event \( A \): Choosing a red card.
- Event \( B \): Choosing an ace.

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

There are 26 red cards in a standard deck (13 diamonds and 13 hearts). Hence, the probability of event \( A \) is:

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

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

There are 4 aces in a standard deck (one from each suit). Thus, the probability of event \( B \) is:

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

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

Event \( A \) and event \( B \) intersect when you choose a red ace. There are 2 red aces in the deck: the Ace of Hearts and the Ace of Diamonds. Therefore:

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

### Step 4: Check Independence of Events A and B

Events A and B are independent if:

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

Calculating \( P(A) \times P(B) \):

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

Since \( P(A \text{ and } B) = P(A) \times P(B) \), we conclude that events A and B are independent.

### Final Answers:

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

Thus, the events A and B are independent.