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
C. mutually exclusive
D. none of these

Answer

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

### Step 1: Write down the given probabilities.
- \( P(A \cap B) = \frac{1}{3} \)
- \( P(A \cup B) = \frac{5}{6} \)
- \( P(A) = \frac{1}{2} \)

### Step 2: Use the formula for the probability of the union of two events.
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 into the formula:
\[ \frac{5}{6} = \frac{1}{2} + P(B) - \frac{1}{3} \]

### Step 3: Solve for \( P(B) \).
First, we need to express all fractions with a common denominator. The least common multiple of 2, 3, and 6 is 6. Rewriting the fractions:
- \( P(A) = \frac{1}{2} = \frac{3}{6} \)
- \( P(A \cap B) = \frac{1}{3} = \frac{2}{6} \)

Now substitute these into the equation:
\[ \frac{5}{6} = \frac{3}{6} + P(B) - \frac{2}{6} \]
This simplifies to:
\[ \frac{5}{6} = \frac{1}{6} + P(B) \]
Now, isolate \( P(B) \):
\[ P(B) = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \]

### Step 4: Check for independence.
For events A and B to be independent, the following must hold true:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Substituting the values we have:
\[ \frac{1}{3} = \left(\frac{1}{2}\right) \cdot \left(\frac{2}{3}\right) \]
Calculating the right side:
\[ \frac{1}{2} \cdot \frac{2}{3} = \frac{2}{6} = \frac{1}{3} \]
Since both sides are equal, A and B are independent.

### Step 5: Check for mutual exclusivity.
For events A and B to be mutually exclusive, the following must hold true:
\[ P(A \cap B) = 0 \]
However, we know that:
\[ P(A \cap B) = \frac{1}{3} \neq 0 \]
Thus, A and B are not mutually exclusive.

### Conclusion:
The correct statement is that A and B are independent.