Let A and B be two events such that `Poverline((AcupB))=(1)/(6),P(AcapB)=(1)/(4) and Poverline(A)=(1)/(4)`,where `overline(A)` stands for complement of event A. then , events A and B are

Options

A. independent but not equally likely
B. mutually exculusive and independent
C. equally likely and mutually exclusive
D. equally likely but not independent

Answer

To solve the problem step by step, we will use the provided probabilities and the definitions of union, intersection, and complements of events.

### Step 1: Identify the given probabilities

We are given:
- \( P(\overline{A \cup B}) = \frac{1}{6} \)
- \( P(A \cap B) = \frac{1}{4} \)
- \( P(\overline{A}) = \frac{1}{4} \)

### Step 2: Calculate \( P(A) \)

Since \( P(\overline{A}) = \frac{1}{4} \), we can find \( P(A) \) using the complement rule:
\[ P(A) = 1 - P(\overline{A}) = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 3: Calculate \( P(A \cup B) \)

Using the complement rule, we can find \( P(A \cup B) \):
\[ P(A \cup B) = 1 - P(\overline{A \cup B}) = 1 - \frac{1}{6} = \frac{5}{6} \]

### Step 4: Use the formula for \( P(A \cup B) \)

The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ \frac{5}{6} = P(A) + P(B) - P(A \cap B) \]
\[ \frac{5}{6} = \frac{3}{4} + P(B) - \frac{1}{4} \]

### Step 5: Simplify the equation

Now, we simplify the equation:
\[ \frac{5}{6} = \frac{3}{4} + P(B) - \frac{1}{4} \]
\[ \frac{5}{6} = \frac{3}{4} - \frac{1}{4} + P(B) \]
\[ \frac{5}{6} = \frac{2}{4} + P(B) \]
\[ \frac{5}{6} = \frac{1}{2} + P(B) \]

### Step 6: Solve for \( P(B) \)

Now, we solve for \( P(B) \):
\[ P(B) = \frac{5}{6} - \frac{1}{2} \]
To subtract, convert \( \frac{1}{2} \) to sixths:
\[ \frac{1}{2} = \frac{3}{6} \]
Thus,
\[ P(B) = \frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3} \]

### Step 7: Check for independence

To check if events A and B are independent, we need to verify if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Substituting the values:
\[ P(A \cap B) = \frac{1}{4}, \quad P(A) = \frac{3}{4}, \quad P(B) = \frac{1}{3} \]
Calculating \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = \frac{3}{4} \cdot \frac{1}{3} = \frac{3}{12} = \frac{1}{4} \]
Since \( P(A \cap B) = P(A) \cdot P(B) \), events A and B are independent.

### Conclusion

Since A and B are independent but not equally likely (as \( P(A) = \frac{3}{4} \) and \( P(B) = \frac{1}{3} \)), the final answer is:

**Events A and B are independent but not equally likely.**

---