Question

For which sets of probabilities are events A and B independent? Select all that apply.

A. P(A)= 0.36, P(B) = 0.4, P(B|A) = 0.4
B. P(A)= 0.7, P(B) = 0.25, P(A and B) = 0.175
C. P(A)= 0.5, P(B) = 0.11, P(A|B) = 0.055
D. P(A)= 0.22, P(B) = 0.48, P(A|B) = 0.48

Answer 1

To determine if two events A and B are independent, we need to check the relationship between their probabilities. Specifically, events A and B are independent if \( P(A \cap B) = P(A) \cdot P(B) \).

We will analyze each option based on the conditions provided.

### Option A:
- \( P(A) = 0.36 \)
- \( P(B) = 0.4 \)
- \( P(B|A) = 0.4 \)

To check for independence, we first calculate \( P(A \cap B) \):
\[
P(A \cap B) = P(B|A) \cdot P(A) = 0.4 \cdot 0.36 = 0.144
\]
Now calculate \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) = 0.36 \cdot 0.4 = 0.144
\]
Since \( P(A \cap B) = P(A) \cdot P(B) \), events A and B are independent.

### Option B:
- \( P(A) = 0.7 \)
- \( P(B) = 0.25 \)
- \( P(A \cap B) = 0.175 \)

Now check:
\[
P(A) \cdot P(B) = 0.7 \cdot 0.25 = 0.175
\]
Since \( P(A \cap B) = P(A) \cdot P(B) \), events A and B are independent.

### Option C:
- \( P(A) = 0.5 \)
- \( P(B) = 0.11 \)
- \( P(A|B) = 0.055 \)

First, compute \( P(A \cap B) \):
\[
P(A \cap B) = P(A|B) \cdot P(B) = 0.055 \cdot 0.11 = 0.00605
\]
Now calculate \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) = 0.5 \cdot 0.11 = 0.055
\]
Since \( P(A \cap B) \neq P(A) \cdot P(B) \), events A and B are not independent.

### Option D:
- \( P(A) = 0.22 \)
- \( P(B) = 0.48 \)
- \( P(A|B) = 0.48 \)

Now check:
\[
P(A \cap B) = P(A|B) \cdot P(B) = 0.48 \cdot 0.48 = 0.2304
\]
Calculate \( P(A) \cdot P(B) \):
\[
P(A) \cdot P(B) = 0.22 \cdot 0.48 = 0.1056
\]
Since \( P(A \cap B) \neq P(A) \cdot P(B) \), events A and B are not independent.

### Final Conclusion
The sets of probabilities for which events A and B are independent are:
- **A and B** (options A and B).