Let A and B be two independent events with P(A) = p and P(B) = q. If p and q are the roots of the equation `ax^(2) + bx + c = 0`, then the probability of the occurrence of at least one of the two events is

Options

A. `(b+c)/a`
B. `(b-c)/a`
C. `(-(b+c))/a`
D. `(-b+c)/a`

Answer

To solve the problem, we need to find the probability of the occurrence of at least one of the two independent events A and B, given that the probabilities of the events are the roots of the quadratic equation \( ax^2 + bx + c = 0 \).

### Step-by-Step Solution:

1. **Identify the Roots of the Quadratic Equation**:
The roots of the equation \( ax^2 + bx + c = 0 \) are given as \( p \) and \( q \). According to Vieta's formulas:
- The sum of the roots \( p + q = -\frac{b}{a} \)
- The product of the roots \( pq = \frac{c}{a} \)

2. **Probability of At Least One Event Occurring**:
For two independent events A and B, the probability of at least one of the events occurring can be calculated using the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since A and B are independent, we have:
\[ P(A \cap B) = P(A) \cdot P(B) = pq \]
Therefore, the formula becomes:
\[ P(A \cup B) = P(A) + P(B) - P(A)P(B) \]
Substituting \( P(A) = p \) and \( P(B) = q \):
\[ P(A \cup B) = p + q - pq \]

3. **Substituting the Values**:
Now, we substitute the values of \( p + q \) and \( pq \) from Vieta's formulas:
- \( p + q = -\frac{b}{a} \)
- \( pq = \frac{c}{a} \)

Thus, we can rewrite the probability:
\[ P(A \cup B) = \left(-\frac{b}{a}\right) - \left(\frac{c}{a}\right) \]

4. **Simplifying the Expression**:
Combine the terms:
\[ P(A \cup B) = -\frac{b + c}{a} \]

5. **Final Result**:
Therefore, the probability of the occurrence of at least one of the two events A or B is:
\[ P(A \cup B) = -\frac{b + c}{a} \]