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.