Question

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

A. P(A)= 0.15, P(B) = 0.24, P(A and B) = 0.072
B. P(A)= 0.6, P(B) = 0.3, P(B|A) = 0.18
C. P(A)= 0.2, P(B) = 0.4, P(A|B) = 0.2
D. P (A) = 0.18, P(B) = 0.3 P(A and B) = 0.054

Answer 1

Two events A and B are independent if and only if the probability of their joint occurrence equals the product of their individual probabilities. Mathematically, this condition is expressed as:

\[ P(A \text{ and } B) = P(A) \cdot P(B) \]

Let's evaluate each set of provided probabilities to see if this condition holds true:

### A.
- \( P(A) = 0.15 \)
- \( P(B) = 0.24 \)
- \( P(A \text{ and } B) = 0.072 \)

Check:
\[ P(A) \cdot P(B) = 0.15 \cdot 0.24 = 0.036 \]

Since \( P(A \text{ and } B) \neq P(A) \cdot P(B) \), A is **not independent**.

### B.
- \( P(A) = 0.6 \)
- \( P(B) = 0.3 \)
- \( P(B|A) = 0.18 \)

Using the definition of conditional probability:
\[ P(B|A) = \frac{P(A \text{ and } B)}{P(A)} \]
So:
\[ P(A \text{ and } B) = P(B|A) \cdot P(A) = 0.18 \cdot 0.6 = 0.108 \]

Check independence:
\[ P(A) \cdot P(B) = 0.6 \cdot 0.3 = 0.18 \]

Since \( P(A \text{ and } B) \neq P(A) \cdot P(B) \), B is **not independent**.

### C.
- \( P(A) = 0.2 \)
- \( P(B) = 0.4 \)
- \( P(A|B) = 0.2 \)

Using the definition of conditional probability:
\[ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \]
So:
\[ P(A \text{ and } B) = P(A|B) \cdot P(B) = 0.2 \cdot 0.4 = 0.08 \]

Check independence:
\[ P(A) \cdot P(B) = 0.2 \cdot 0.4 = 0.08 \]

Since \( P(A \text{ and } B) = P(A) \cdot P(B) \), C is **independent**.

### D.
- \( P(A) = 0.18 \)
- \( P(B) = 0.3 \)
- \( P(A \text{ and } B) = 0.054 \)

Check:
\[ P(A) \cdot P(B) = 0.18 \cdot 0.3 = 0.054 \]

Since \( P(A \text{ and } B) = P(A) \cdot P(B) \), D is **independent**.

### Conclusion
The sets of probabilities for which events A and B are independent are:
- **C**
- **D**