Question

Given the following information, can we determine which pairs of A, B, and C are dependent and which are independent?
a) P(A) = 0.4 P(B|A) = 0.7
b) P(B) = 0.3 P(C|B) = 0.2
c) P(C) = 0.2 P(C|A) = 0.3

Answer 1

To determine the dependence and independence of the variables A, B, and C, we can use the relationships given in the probabilities.

1. **Independence between A and B:**
- We know \(P(A) = 0.4\) and \(P(B|A) = 0.7\).
- For A and B to be independent, it must hold that \(P(B|A) = P(B)\).
- Since \(P(B) = 0.3\) (given in b), and \(P(B|A) = 0.7\), this means that \(P(B|A) \neq P(B)\).
- Thus, A and B are **dependent**.

2. **Independence between B and C:**
- We have \(P(B) = 0.3\) and \(P(C|B) = 0.2\).
- For B and C to be independent, we need \(P(C|B) = P(C)\).
- Since \(P(C) = 0.2\) (given in c) and \(P(C|B) = 0.2\), we find that \(P(C|B) = P(C)\).
- Therefore, B and C are **independent**.

3. **Independence between C and A:**
- Here, we know \(P(C) = 0.2\) and \(P(C|A) = 0.3\).
- For C and A to be independent, it must hold that \(P(C|A) = P(C)\).
- Since \(P(C|A) = 0.3\) and \(P(C) = 0.2\), we have \(P(C|A) \neq P(C)\).
- Thus, C and A are **dependent**.

In summary:
- A and B are **dependent**.
- B and C are **independent**.
- A and C are **dependent**.