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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 calculate the probabilities of the events defined.

  1. P(A and B): This is the probability of drawing a card that is both a red card and an ace.

    • There are 2 red aces in the deck: the Ace of Hearts and the Ace of Diamonds.
    • Therefore, \( P(A \text{ and } B) = \frac{\text{Number of red aces}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26} \).

  2. P(A): This is the probability of drawing a red card.

    • There are a total of 26 red cards (13 diamonds and 13 hearts).
    • Therefore, \( P(A) = \frac{\text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2} \).

  3. P(B): This is the probability of drawing an ace.

    • There are 4 aces in the deck (one for each suit).
    • Therefore, \( P(B) = \frac{\text{Number of aces}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13} \).

Now we state whether events A and B are independent. Events A and B are independent if \( P(A \text{ and } B) = P(A) \times P(B) \).

Now let's check this: \[ P(A) \times P(B) = \left(\frac{1}{2}\right) \times \left(\frac{1}{13}\right) = \frac{1}{26} \] Since \( P(A \text{ and } B) = \frac{1}{26} \) and \( P(A) \times P(B) = \frac{1}{26} \), it follows that the events A and B are independent.

To summarize:

  • \( P(A \text{ and } B) = \frac{1}{26} \)
  • \( P(A) = \frac{1}{2} \)
  • \( P(B) = \frac{1}{13} \)
  • Since \( P(A \text{ and } B) = P(A) \times P(B) \), events A and B are independent.

Final answers:

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