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
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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
Skip to navigation
Answers
GPT-4o mini
Let's start by calculating \( P(A) \), \( P(B) \), and then determine \( P(A) \cdot P(B) \) and \( P(A \cap B) \).
1. **Calculating \( P(A) \)**: Event A is choosing a black card. There are 26 black cards (13 spades and 13 clubs) in a standard deck of 52 cards.
\[
P(A) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}
\]
2. **Calculating \( P(B) \)**: Event B is choosing a king. There are 4 kings in a standard deck of 52 cards.
\[
P(B) = \frac{\text{Number of kings}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}
\]
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 \( P(A \cap B) \)**: \( P(A \cap B) \) is the probability of choosing a card that is both black and a king. There are 2 black kings (the king of spades and the king of clubs). So:
\[
P(A \cap B) = \frac{\text{Number of black kings}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}
\]
5. **Determining if events A and B are independent**: Two events A and B are independent if \( P(A \cap B) = P(A) \cdot P(B) \).
We found:
\[
P(A) \cdot P(B) = \frac{1}{26}
\]
and
\[
P(A \cap B) = \frac{1}{26}
\]
Since \( P(A \cap B) = P(A) \cdot P(B) \), events A and B are independent.
### Final Answers:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \cap B) = \frac{1}{26} \)
- Events A and B are independent (value = 2).
Thus, the complete answers are:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \cap B) = \frac{1}{26} \)
- Events A and B are independent: **2**.
1. **Calculating \( P(A) \)**: Event A is choosing a black card. There are 26 black cards (13 spades and 13 clubs) in a standard deck of 52 cards.
\[
P(A) = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}
\]
2. **Calculating \( P(B) \)**: Event B is choosing a king. There are 4 kings in a standard deck of 52 cards.
\[
P(B) = \frac{\text{Number of kings}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13}
\]
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 \( P(A \cap B) \)**: \( P(A \cap B) \) is the probability of choosing a card that is both black and a king. There are 2 black kings (the king of spades and the king of clubs). So:
\[
P(A \cap B) = \frac{\text{Number of black kings}}{\text{Total number of cards}} = \frac{2}{52} = \frac{1}{26}
\]
5. **Determining if events A and B are independent**: Two events A and B are independent if \( P(A \cap B) = P(A) \cdot P(B) \).
We found:
\[
P(A) \cdot P(B) = \frac{1}{26}
\]
and
\[
P(A \cap B) = \frac{1}{26}
\]
Since \( P(A \cap B) = P(A) \cdot P(B) \), events A and B are independent.
### Final Answers:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \cap B) = \frac{1}{26} \)
- Events A and B are independent (value = 2).
Thus, the complete answers are:
- \( P(A) \cdot P(B) = \frac{1}{26} \)
- \( P(A \cap B) = \frac{1}{26} \)
- Events A and B are independent: **2**.