Let A and B be two events such that P(A|B)=1/2 , P(B|A)=1/3, `P(A nn B)`=1/6 , then

Options

A. `P(A uu B)=1//2`
B. A and B are independent
C. `P(A' uu B)`=1/3
D. none of these

Answer

To solve the problem, we need to find the probabilities of events A and B using the given information. We have:

1. \( P(A|B) = \frac{1}{2} \)
2. \( P(B|A) = \frac{1}{3} \)
3. \( P(A \cap B) = \frac{1}{6} \)

### Step 1: Use the definition of conditional probability

The formula for conditional probability is given by:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Substituting the known values:

\[ \frac{1}{2} = \frac{\frac{1}{6}}{P(B)} \]

### Step 2: Solve for \( P(B) \)

Cross-multiplying gives:

\[ P(B) = \frac{1}{6} \times 2 = \frac{1}{3} \]

### Step 3: Use the second conditional probability

Now, using the second conditional probability:

\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

Substituting the known values:

\[ \frac{1}{3} = \frac{\frac{1}{6}}{P(A)} \]

### Step 4: Solve for \( P(A) \)

Cross-multiplying gives:

\[ P(A) = \frac{1}{6} \times 3 = \frac{1}{2} \]

### Step 5: Calculate \( P(A \cup B) \)

We can now find \( P(A \cup B) \) using the formula:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substituting the values we found:

\[ P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} \]

### Step 6: Find a common denominator and simplify

The common denominator for \( \frac{1}{2}, \frac{1}{3}, \frac{1}{6} \) is 6. Thus:

\[ P(A \cup B) = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \]

### Step 7: Check for independence

To check if A and B are independent, we need to see if:

\[ P(A \cap B) = P(A) \cdot P(B) \]

Substituting the values we found:

\[ \frac{1}{6} = \frac{1}{2} \cdot \frac{1}{3} \]

Calculating the right side:

\[ \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \]

Since both sides are equal, A and B are independent.

### Conclusion

Thus, we have found:

- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{3} \)
- \( P(A \cup B) = \frac{2}{3} \)
- A and B are independent.