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

Question

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).