Question

A standard deck of cards has 52 cards. You choose one card from the deck. Let event A be you choose a black card and event B be you choose a king. Find P(A)⋅P(B) , P(A and B) , and determine whether events A and B are independent.

Enter the first two answers in fraction form.

(2 points)
P(A)⋅P(B)=
;
P(A and 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 \( P(A) \) and \( P(B) \), and determine the independence of the events, we start with identifying the events and their outcomes in a standard deck of 52 cards.

**Event A:** You choose a black card.
There are 26 black cards in a standard deck (13 spades + 13 clubs).
Thus, the probability of selecting a black card is:
\[
P(A) = \frac{26}{52} = \frac{1}{2}
\]

**Event B:** You choose a king.
There are 4 kings in the deck (one in each suit).
Thus, the probability of selecting a king is:
\[
P(B) = \frac{4}{52} = \frac{1}{13}
\]

Now we can calculate \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{13}\right) = \frac{1}{26}
\]

Next, we find \( P(A \text{ and } B) \), which represents the probability of selecting a card that is both black and a king.
There are 2 black kings (King of Spades and King of Clubs) in the deck.
Thus,
\[
P(A \text{ and } B) = \frac{2}{52} = \frac{1}{26}
\]

Now we check 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 already have:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \text{ and } B) = \frac{1}{26} \)

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

Putting it all together, we have:

\[
P(A) \cdot P(B) = \frac{1}{26}
\]
\[
P(A \text{ and } B) = \frac{1}{26}
\]
Events A and B are independent, so we enter `2` as our final answer.

Thus, the final answers are:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \text{ and } B) = \frac{1}{26} \)
- Events A and B are independent: `2`