If `P(A cap B)=1//3, P(A cup B)=5//6 " and " P(A)=1//2`, then which one of the following is correct ?

Options

A. A and B are independent events
B. A and B are mutually exclusive events
C. P(A)=P(B)
D. `P(A) lt P(B)`

Explanation

To solve the problem, we will use the given probabilities and the properties of probability.

### Given:
- \( P(A \cap B) = \frac{1}{3} \)
- \( P(A \cup B) = \frac{5}{6} \)
- \( P(A) = \frac{1}{2} \)

### Step 1: Use the formula for the probability of the union of two events.
The formula for the probability of the union of two events is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 2: Substitute the known values into the formula.
Substituting the known values into the formula, we have:
\[ \frac{5}{6} = \frac{1}{2} + P(B) - \frac{1}{3} \]

### Step 3: Simplify the equation to find \( P(B) \).
First, convert all fractions to have a common denominator. The least common multiple of 6 is 6. Therefore, we rewrite the fractions:
- \( \frac{1}{2} = \frac{3}{6} \)
- \( \frac{1}{3} = \frac{2}{6} \)

Now substitute these into the equation:
\[ \frac{5}{6} = \frac{3}{6} + P(B) - \frac{2}{6} \]

### Step 4: Combine the fractions on the right side.
\[ \frac{5}{6} = \frac{3}{6} - \frac{2}{6} + P(B) \]
\[ \frac{5}{6} = \frac{1}{6} + P(B) \]

### Step 5: Solve for \( P(B) \).
To isolate \( P(B) \), subtract \( \frac{1}{6} \) from both sides:
\[ P(B) = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \]

### Step 6: Check the independence of events A and B.
To check if events A and B are independent, we need to verify if:
\[ P(A \cap B) = P(A) \times P(B) \]

Substituting the values we have:
\[ P(A) = \frac{1}{2}, \quad P(B) = \frac{2}{3} \]
Calculating \( P(A) \times P(B) \):
\[ P(A) \times P(B) = \frac{1}{2} \times \frac{2}{3} = \frac{1 \cdot 2}{2 \cdot 3} = \frac{2}{6} = \frac{1}{3} \]

### Step 7: Compare with \( P(A \cap B) \).
Since \( P(A \cap B) = \frac{1}{3} \), we find that:
\[ P(A \cap B) = P(A) \times P(B) \]
This confirms that A and B are independent events.

### Conclusion:
- \( P(B) = \frac{2}{3} \)
- Events A and B are independent.

### Final Answer:
- Option 1: A and B are independent (Correct)
- Option 4: \( P(B) > P(A) \) (Correct)