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 Remaining Attempts : 3
Answers
GPT-4o mini
To solve the problem, we will first find \( P(A) \), \( P(B) \), and \( P(A \text{ and } B) \).
**Step 1: Calculate \( P(A) \)**
Event A is choosing a black card. There are 26 black cards in a standard deck (13 spades and 13 clubs).
\[
P(A) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}
\]
**Step 2: Calculate \( P(B) \)**
Event B is choosing a king. There are 4 kings in a deck (one from each suit).
\[
P(B) = \frac{\text{Number of kings}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}
\]
**Step 3: Calculate \( P(A \text{ and } B) \)**
Event A and B happening together (choosing a card that is both black and a king) would mean choosing either the king of spades or the king of clubs, which gives us 2 favorable outcomes.
\[
P(A \text{ and } B) = \frac{\text{Number of black kings}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}
\]
**Step 4: Calculate \( P(A) \cdot P(B) \)**
\[
P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{13} = \frac{1}{26}
\]
**Step 5: Determine Independence**
Two events A and B are independent if:
\[
P(A \text{ and } B) = P(A) \cdot P(B)
\]
We see that:
\[
P(A \text{ and } B) = \frac{1}{26}
\]
\[
P(A) \cdot P(B) = \frac{1}{26}
\]
Since these values are equal, events A and B are independent.
**Final Answers:**
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \text{ and } B) = \frac{1}{26} \)
- Events A and B are independent, so enter 2.
Thus,
\[
\text{P(A)⋅P(B)} = \frac{1}{26}, \quad \text{P(A and B)} = \frac{1}{26}, \quad \text{independence} = 2
\]
**Step 1: Calculate \( P(A) \)**
Event A is choosing a black card. There are 26 black cards in a standard deck (13 spades and 13 clubs).
\[
P(A) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}
\]
**Step 2: Calculate \( P(B) \)**
Event B is choosing a king. There are 4 kings in a deck (one from each suit).
\[
P(B) = \frac{\text{Number of kings}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}
\]
**Step 3: Calculate \( P(A \text{ and } B) \)**
Event A and B happening together (choosing a card that is both black and a king) would mean choosing either the king of spades or the king of clubs, which gives us 2 favorable outcomes.
\[
P(A \text{ and } B) = \frac{\text{Number of black kings}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}
\]
**Step 4: Calculate \( P(A) \cdot P(B) \)**
\[
P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{13} = \frac{1}{26}
\]
**Step 5: Determine Independence**
Two events A and B are independent if:
\[
P(A \text{ and } B) = P(A) \cdot P(B)
\]
We see that:
\[
P(A \text{ and } B) = \frac{1}{26}
\]
\[
P(A) \cdot P(B) = \frac{1}{26}
\]
Since these values are equal, events A and B are independent.
**Final Answers:**
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \text{ and } B) = \frac{1}{26} \)
- Events A and B are independent, so enter 2.
Thus,
\[
\text{P(A)⋅P(B)} = \frac{1}{26}, \quad \text{P(A and B)} = \frac{1}{26}, \quad \text{independence} = 2
\]