Question

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

Answer 1

To calculate the probabilities, we first need to determine the number of red cards and aces in the deck:

- There are 26 red cards in a standard deck (13 diamonds and 13 hearts).
- There are 4 aces in a standard deck (one in each suit).

1. P(A and B): Event A and B both happening means choosing a red ace. There are 2 red aces in the deck: the ace of diamonds and the ace of hearts. Therefore, P(A and B) = 2/52 = 1/26.

2. P(A): The probability of choosing a red card is the number of red cards in the deck divided by the total number of cards: 26/52 = 1/2.

3. P(B): The probability of choosing an ace is the number of aces in the deck divided by the total number of cards: 4/52 = 1/13.

To determine if events A and B are independent, we need to check if P(A and B) = P(A) * P(B). Let's calculate:

P(A) * P(B) = (1/2) * (1/13) = 1/26

Since P(A and B) = 1/26 is not equal to P(A) * P(B) = 1/26, events A and B are dependent (answer: 1).