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.

Answer 1

To solve this problem, we first need to determine the probabilities of events A and B, and then calculate \( P(A \cap B) \), the probability of both events occurring.

1. **Calculating \( P(A) \)**:
- Event A: Choosing a black card.
- In a standard deck of 52 cards, there are 26 black cards (13 clubs and 13 spades).
- Therefore,
\[
P(A) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}.
\]

2. **Calculating \( P(B) \)**:
- Event B: Choosing a king.
- In a standard deck, there are 4 kings (one from each suit).
- Thus,
\[
P(B) = \frac{\text{Number of kings}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}.
\]

3. **Calculating \( P(A \cap B) \)**:
- Event \( A \cap B \): Choosing a card that is both black and a king.
- There are 2 black kings (the king of clubs and the king of spades).
- Therefore,
\[
P(A \cap B) = \frac{\text{Number of black kings}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}.
\]

4. **Calculating \( P(A) \cdot P(B) \)**:
- Now we can calculate \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{13}\right) = \frac{1}{26}.
\]

5. **Determining independence**:
- Events A and B are independent if
\[
P(A \cap B) = P(A) \cdot P(B).
\]
- We calculated:
\[
P(A \cap B) = \frac{1}{26} \text{ and } P(A) \cdot P(B) = \frac{1}{26}.
\]
- Since these are equal, events A and B are independent.

### Summary of Results:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \cap B) = \frac{1}{26} \)
- Events A and B are **independent**.