If A and B are two events such that `P[(A uu B)']=1/6, P(A nn B)=1/4` and `P(A')=1/4` , then events A and B are

Options

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

Answer

To solve the problem, we need to find the probabilities of events A and B based on the given information. Let's go through the steps systematically.

### Given:
1. \( P[(A \cup B)'] = \frac{1}{6} \)
2. \( P(A \cap B) = \frac{1}{4} \)
3. \( P(A') = \frac{1}{4} \)

### Step 1: Find \( P(A \cup B) \)
Using the complement rule:
\[ P(A \cup B) = 1 - P[(A \cup B)'] = 1 - \frac{1}{6} = \frac{5}{6} \]

### Step 2: Find \( P(A) \)
Since \( P(A') = \frac{1}{4} \), we can find \( P(A) \):
\[ P(A) = 1 - P(A') = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 3: Use the formula for \( P(A \cup B) \)
We know that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the values we have:
\[ \frac{5}{6} = P(A) + P(B) - P(A \cap B) \]
Substituting \( P(A) = \frac{3}{4} \) and \( P(A \cap B) = \frac{1}{4} \):
\[ \frac{5}{6} = \frac{3}{4} + P(B) - \frac{1}{4} \]
This simplifies to:
\[ \frac{5}{6} = \frac{3}{4} - \frac{1}{4} + P(B) \]
\[ \frac{5}{6} = \frac{2}{4} + P(B) = \frac{1}{2} + P(B) \]

### Step 4: Solve for \( P(B) \)
Now, we need to isolate \( P(B) \):
\[ P(B) = \frac{5}{6} - \frac{1}{2} \]
To subtract these fractions, we need a common denominator:
\[ P(B) = \frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3} \]

### Step 5: Check for independence
To check if A and B are independent, we need to verify if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Substituting the values we found:
\[ \frac{1}{4} = \left(\frac{3}{4}\right) \cdot \left(\frac{1}{3}\right) \]
Calculating the right side:
\[ \frac{3}{4} \cdot \frac{1}{3} = \frac{3}{12} = \frac{1}{4} \]
Since both sides are equal, A and B are independent.

### Conclusion
Events A and B are independent but not equally likely.