If `(A) =1/3 ,P(B) =1/2 and P(A cup B) =5/6` then events A and B are

Options

A. independent
B. independent and mutually exclusive
C. mutually exclusive
D. none of these

Answer

To determine the relationship between events A and B given the probabilities \( P(A) = \frac{1}{3} \), \( P(B) = \frac{1}{2} \), and \( P(A \cup B) = \frac{5}{6} \), we will follow these steps:

### 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:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
We can rearrange this to find \( P(A \cap B) \):
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]

### Step 2: Substitute the known values into the formula
Substituting the given probabilities into the equation:
\[ P(A \cap B) = \frac{1}{3} + \frac{1}{2} - \frac{5}{6} \]

### Step 3: Find a common denominator and calculate
The least common multiple of the denominators (3, 2, and 6) is 6. We convert each fraction:
- \( \frac{1}{3} = \frac{2}{6} \)
- \( \frac{1}{2} = \frac{3}{6} \)
- \( \frac{5}{6} = \frac{5}{6} \)

Now substituting these values:
\[ P(A \cap B) = \frac{2}{6} + \frac{3}{6} - \frac{5}{6} \]
\[ P(A \cap B) = \frac{2 + 3 - 5}{6} = \frac{0}{6} = 0 \]

### Step 4: Analyze the result
Since \( P(A \cap B) = 0 \), this means that events A and B do not occur together.

### Step 5: Determine if A and B are independent or mutually exclusive
1. **Independent Events**: Two events A and B are independent if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Here, \( P(A) \cdot P(B) = \frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6} \), which is not equal to \( P(A \cap B) = 0 \). Therefore, A and B are not independent.

2. **Mutually Exclusive Events**: Two events A and B are mutually exclusive if:
\[ P(A \cap B) = 0 \]
Since we found that \( P(A \cap B) = 0 \), A and B are mutually exclusive.

### Conclusion
Events A and B are **mutually exclusive**.

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