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A standard deck of cards has 52 cards. The 52 cards are divided into four suits: diamonds, clubs, hearts, and spades. Diamonds and hearts are red cards, and clubs and spades are black cards. Suppose you choose a card from the deck. Let event A

be you choose a red card and event B
you choose an ace. Determine P(A and B)
, P(A)
, and P(B)
, and state whether events A
and B
are independent.

Enter the first three answers in fraction form. Reduce your answer.

(2 points)
P(A and B)=
;
P(A)=
;
P(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 need to determine the probabilities for the events A and B, as well as whether they are independent.

  1. Determine \( P(A \text{ and } B) \):

    • Event A (choosing a red card): There are 26 red cards in total (13 diamonds + 13 hearts).
    • Event B (choosing an ace): There are 4 aces in total (one for each suit).
    • The only red aces are the Ace of Diamonds and the Ace of Hearts, so there are 2 outcomes that satisfy both events A and B.

    Therefore, \[ P(A \text{ and } B) = \frac{\text{Number of outcomes in } A \text{ and } B}{\text{Total number of outcomes}} = \frac{2}{52} = \frac{1}{26} \]

  2. Determine \( P(A) \):

    • The probability of event A (choosing a red card): \[ P(A) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2} \]

  3. Determine \( P(B) \):

    • The probability of event B (choosing an ace): \[ P(B) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13} \]

  4. Determine if events A and B are independent:

    • Two events A and B are independent if \[ P(A \text{ and } B) = P(A) \cdot P(B) \]
    • We calculated:
      • \( P(A \text{ and } B) = \frac{1}{26} \)
      • \( P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{13} = \frac{1}{26} \)

Since \( P(A \text{ and } B) = P(A) \cdot P(B) \), we conclude that events A and B are independent.

Final Answers:

  • \( P(A \text{ and } B) = \frac{1}{26} \)
  • \( P(A) = \frac{1}{2} \)
  • \( P(B) = \frac{1}{13} \)
  • Events A and B are independent (enter 2).

So, the responses are:

  • \( P(A \text{ and } B) = \frac{1}{26} \)
  • \( P(A) = \frac{1}{2} \)
  • \( P(B) = \frac{1}{13} \)
  • Enter 2 for independence.