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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.



Question 1
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

1 answer

Let's analyze the problem using the standard deck of cards.

Question 1

Event A: Choosing an ace. In a standard deck, there are 4 aces (one for each suit: hearts, diamonds, clubs, spades).

Event B: Choosing a red card. There are 26 red cards in a standard deck (13 diamonds and 13 hearts).

1. P(A ∩ B): This is the probability of choosing a card that is both an ace and a red card. The only red aces are the Ace of Diamonds and the Ace of Hearts, so there are 2 cards that satisfy both criteria.

\[ P(A \cap B) = \frac{\text{Number of favorable outcomes for } A \cap B}{\text{Total number of outcomes}} = \frac{2}{52} = \frac{1}{26} \]

2. P(A): The probability of choosing an ace.

\[ P(A) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13} \]

3. P(B): The probability of choosing a red card.

\[ P(B) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2} \]

Summary for Question 1:

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

Question 2

Are events A and B dependent?

Events A and B are dependent if the occurrence of one affects the probability of the other occurring.

  • If you pick a card and it's an ace (event A), it would change the composition of the deck, specifically affecting the probability of drawing a red card (event B).
  • The probability of drawing a red card does change if you draw an ace (if you drew the Ace of Hearts or Ace of Diamonds, it would decrease the number of red cards).

Thus, the response is True.

Summary for Question 2:

  • True: In the given situation, events A and B are dependent.