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).
3 answers