Probability Unit Test

Question

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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 solve for the probabilities of events \(A\) and \(B\) and determine their independence, we will follow these steps:

1. Identify the events:

   • Event \(A\) is choosing a red card (diamonds or hearts).
   • Event \(B\) is choosing an ace.

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

   • There are 26 red cards in a standard deck (13 diamonds + 13 hearts).
   • The total number of cards is 52. \[ P(A) = \frac{26}{52} = \frac{1}{2} \]

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

   • There are 4 aces in a standard deck (one for each suit). \[ P(B) = \frac{4}{52} = \frac{1}{13} \]

4. Calculate \(P(A \text{ and } B)\):

   • For event \(A \text{ and } B\) (choosing a red card that is also an ace), the relevant aces are the Ace of Diamonds and the Ace of Hearts. There are 2 red aces. \[ P(A \text{ and } B) = \frac{2}{52} = \frac{1}{26} \]

5. Determine independence:

   • To check if events \(A\) and \(B\) are independent, we need to see if: \[ P(A \text{ and } B) = P(A) \times P(B) \]
   • Calculate \(P(A) \times P(B)\): \[ P(A) \times P(B) = \frac{1}{2} \times \frac{1}{13} = \frac{1}{26} \]
   • Since \(P(A \text{ and } B) = P(A) \times P(B)\), events \(A\) and \(B\) are independent.

Putting it all together:

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

Final answers:

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