Ask a New Question
Menu

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)= ; P(A and B)= ; Enter 1 if events A and B are dependent. Enter 2 if events A and B are independent

1 answer

To solve this problem, we first need to calculate the probabilities for the events A and B.

  1. Event A: Choosing a black card.

    • There are 26 black cards in a standard 52-card deck (13 spades and 13 clubs).
    • Therefore, \( P(A) = \frac{26}{52} = \frac{1}{2} \).

  2. Event B: Choosing a king.

    • There are 4 kings in a standard 52-card deck (one from each suit).
    • Therefore, \( P(B) = \frac{4}{52} = \frac{1}{13} \).

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

  4. Finding \( P(A \text{ and } B) \):

    • The event \( A \text{ and } B \) corresponds to choosing a black king. There are 2 black kings (the king of spades and the king of clubs).
    • Therefore, \( P(A \text{ and } B) = \frac{2}{52} = \frac{1}{26} \).

  5. Determining if A and B are independent:

    • Two events A and B are independent if \( P(A \text{ and } B) = P(A) \cdot P(B) \).
    • Here, \( P(A \text{ and } B) = \frac{1}{26} \) and \( P(A) \cdot P(B) = \frac{1}{26} \).
    • Since these are equal, events A and B are independent.

Putting it all together, we have:

  • \( P(A) \cdot P(B) = \frac{1}{26} \)
  • \( P(A \text{ and } B) = \frac{1}{26} \)
  • Events A and B are independent (enter 2).

Final answers:

  • \( P(A) \cdot P(B) = \frac{1}{26} \)
  • \( P(A \text{ and } B) = \frac{1}{26} \)
  • Independence of events: 2