If P(A) `gt` 0, then the event A is independent of itself if and only if P(A) is

Options

A. `1/3`
B. `1/2`
C. `1`
D. `1/4`

Answer

To solve the problem, we need to determine the conditions under which an event A is independent of itself, given that \( P(A) > 0 \).

### Step-by-Step Solution:

1. **Understanding Independence**:
An event A is independent of itself if the probability of the intersection of A with itself equals the product of their probabilities. Mathematically, this is expressed as:
\[ P(A \cap A) = P(A) \cdot P(A) \]

2. **Simplifying the Intersection**:
Since \( A \cap A \) is simply A, we can rewrite the equation:
\[ P(A) = P(A) \cdot P(A) \]

3. **Setting Up the Equation**:
Let \( P(A) = p \). The equation now looks like:
\[ p = p \cdot p \]
This simplifies to:
\[ p = p^2 \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ p^2 - p = 0 \]
Factoring out p, we have:
\[ p(p - 1) = 0 \]

5. **Finding the Solutions**:
The solutions to this equation are:
\[ p = 0 \quad \text{or} \quad p = 1 \]

6. **Considering the Given Condition**:
Since we are given that \( P(A) > 0 \), we can discard the solution \( p = 0 \). Therefore, the only valid solution is:
\[ P(A) = 1 \]

### Conclusion:
Thus, the event A is independent of itself if and only if \( P(A) = 1 \).