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 this problem, first calculate \( P(A) \) and \( P(B) \).

1. **Calculating \( P(A) \)**:
- The number of black cards in a standard deck (clubs and spades) is 26.
- Therefore, \( P(A) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2} \).

2. **Calculating \( P(B) \)**:
- The number of kings in a standard deck is 4 (one for each suit).
- Therefore, \( P(B) = \frac{\text{Number of kings}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13} \).

3. **Calculating \( P(A) \cdot P(B) \)**:
\[
P(A) \cdot P(B) = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{13}\right) = \frac{1}{26}.
\]

4. **Calculating \( P(A \text{ and } B) \)**:
- For \( P(A \text{ and } B) \), we need to find the probability of drawing a black king. There are 2 black kings (King of Spades and King of Clubs).
- Therefore, \( P(A \text{ and } B) = \frac{\text{Number of black kings}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26} \).

5. **Checking independence**:
- Two events A and B are independent if \( P(A \text{ and } B) = P(A) \cdot P(B) \).
- We see that:
\[
P(A \text{ and } B) = \frac{1}{26}
\]
and
\[
P(A) \cdot P(B) = \frac{1}{26}.
\]
Since these two probabilities are equal, events A and B are independent.

Putting it all together, the answers are:

1. \( P(A) \cdot P(B) = \frac{1}{26} \)
2. \( P(A \text{ and } B) = \frac{1}{26} \)
3. Events A and B are independent: **2**.