Check whether the following probabilities P(A) and P(B) the consistently defined:
(i) `P(A)=0.5, P(B)=0.7, P(AnnB)=0.06 `
(ii) `P(A)=0.5, P(B)=0.4, P(AuuB)=0.8`

Answer

To determine whether the probabilities \( P(A) \) and \( P(B) \) are consistently defined, we need to check the following conditions for both parts of the question:

1. For part (i), we need to check:
- \( P(A \cap B) < P(A) \)
- \( P(A \cap B) < P(B) \)

2. For part (ii), since we are given \( P(A \cup B) \), we need to find \( P(A \cap B) \) using the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Then we will check:
- \( P(A \cap B) < P(A) \)
- \( P(A \cap B) < P(B) \)

Now, let's solve each part step by step.

### Part (i)
Given:
- \( P(A) = 0.5 \)
- \( P(B) = 0.7 \)
- \( P(A \cap B) = 0.06 \)

**Step 1:** Check if \( P(A \cap B) < P(A) \):
\[ 0.06 < 0.5 \quad \text{(True)} \]

**Step 2:** Check if \( P(A \cap B) < P(B) \):
\[ 0.06 < 0.7 \quad \text{(True)} \]

Since both conditions are satisfied, we conclude that \( P(A) \) and \( P(B) \) are consistently defined in part (i).

### Part (ii)
Given:
- \( P(A) = 0.5 \)
- \( P(B) = 0.4 \)
- \( P(A \cup B) = 0.8 \)

**Step 1:** Use the formula to find \( P(A \cap B) \):
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the values:
\[ 0.8 = 0.5 + 0.4 - P(A \cap B) \]
\[ 0.8 = 0.9 - P(A \cap B) \]
Rearranging gives:
\[ P(A \cap B) = 0.9 - 0.8 = 0.1 \]

**Step 2:** Check if \( P(A \cap B) < P(A) \):
\[ 0.1 < 0.5 \quad \text{(True)} \]

**Step 3:** Check if \( P(A \cap B) < P(B) \):
\[ 0.1 < 0.4 \quad \text{(True)} \]

Since both conditions are satisfied, we conclude that \( P(A) \) and \( P(B) \) are consistently defined in part (ii) as well.

### Final Conclusion
- For part (i): \( P(A) \) and \( P(B) \) are consistently defined.
- For part (ii): \( P(A) \) and \( P(B) \) are consistently defined.