Question

A standard deck of cards has 52 cards. You choose one card from the deck. Let event A be you choose a black card and event B be you choose a king. Find P(A)⋅P(B) , P(A and B) , and determine whether events A and B are independent. Enter the first two answers in fraction form. (2 points) P(A)⋅P(B)=

Answer 1

To solve for \( P(A) \), \( P(B) \), and their product \( P(A) \cdot P(B) \), as well as \( P(A \text{ and } B) \) and whether they are independent, we proceed as follows:

1. **Determine \( P(A) \)**:
- There are a total of 52 cards in a standard deck.
- The black cards in a deck are the clubs and spades. There are 26 black cards.
\[
P(A) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}
\]

2. **Determine \( P(B) \)**:
- There are 4 kings in a deck of 52 cards (one for each suit).
\[
P(B) = \frac{\text{Number of kings}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}
\]

3. **Calculate \( P(A) \cdot P(B) \)**:
\[
P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{13} = \frac{1}{26}
\]

4. **Determine \( P(A \text{ and } B) \)**:
- The event \( A \text{ and } B \) corresponds to picking a black king. There are 2 black kings (the King of Spades and the King of Clubs).
\[
P(A \text{ and } B) = \frac{\text{Number of black kings}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}
\]

5. **Check for Independence**:
- Two events \( A \) and \( B \) are independent if \( P(A \text{ and } B) = P(A) \cdot P(B) \).
- From our calculations:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \text{ and } B) = \frac{1}{26} \)

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

Thus, the final answers are:

1. \( P(A) \cdot P(B) = \frac{1}{26} \)
2. \( P(A \text{ and } B) = \frac{1}{26} \)

And the events \( A \) and \( B \) are independent.