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A standard deck of cards has 52 card. The 52 cards are divided into 4 units: diamonds, clubs, hearts, and spades. The diamond and the hearts are red cards and the clubs and the spades are black. See the picture below for an example. For the following problems, consider the following scenario: Brant is doing a card trick and asks you to pick a card from a standard deck. Let event A be that you choose an ace and let event B be that you choose a red card.

Determine the following . Enter your answer as a fraction(3 points)

P (A ∩ B) =

P(A) =

P(B) =

Question 2 -

True or False: In the given situation, events A and B are dependent.(1 point) Responses True True False False

1 answer

To solve the probability questions about events A and B concerning a standard deck of cards, let's define both events clearly:

  • Event A: Choosing an ace from the deck.
  • Event B: Choosing a red card from the deck.

Now, let's calculate the probabilities.

  1. P(A ∩ B):

    • The intersection \(A ∩ B\) represents the probability of choosing an ace that is also a red card. In a standard deck, there are 2 red aces: the Ace of Diamonds and the Ace of Hearts.
    • Thus, \(P(A ∩ B) = \frac{\text{Number of red aces}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}\).

  2. P(A):

    • The event A consists of all 4 aces in the deck: Ace of Diamonds, Ace of Hearts, Ace of Clubs, and Ace of Spades.
    • Thus, \(P(A) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}\).

  3. P(B):

    • The event B consists of all red cards, which include the 26 red cards (13 Diamonds and 13 Hearts).
    • Thus, \(P(B) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}\).

Now, we can summarize the answers:

  • P(A ∩ B) = \(\frac{1}{26}\)
  • P(A) = \(\frac{1}{13}\)
  • P(B) = \(\frac{1}{2}\)

Question 2:

For the second question, we need to determine if events A and B are dependent or independent. Events are dependent if the occurrence of one affects the occurrence of the other.

  • In this scenario, knowing that a card chosen is red (event B has occurred) does affect the probability of drawing an ace (event A). Specifically, there are only 2 aces that are red. If we draw and know the card is red, we reduce the possibilities for drawing an ace, which makes the events dependent.

Therefore, the statement "events A and B are dependent" is True.

Summary:

  • P(A ∩ B) = \(\frac{1}{26}\)
  • P(A) = \(\frac{1}{13}\)
  • P(B) = \(\frac{1}{2}\)
  • Events A and B are dependent: True.