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