Given two independent events, if the probability that exactly one of them occurs is 26/49 and the probability that none of them occurs is 15/49, then the probability of more probable of two events is

Options

A. `(4)/(7)`
B. `(6)/(7)`
C. `(3)/(7)`
D. `(5)/(7)`

Answer

To solve the problem, we need to find the probabilities of two independent events A and B, given the probabilities of exactly one of them occurring and none of them occurring.

### Step-by-Step Solution:

1. **Define the Probabilities:**
Let \( P(A) = a \) and \( P(B) = b \). The probabilities of not occurring are \( P(A') = 1 - a \) and \( P(B') = 1 - b \).

2. **Use the Given Information:**
We know:
- The probability that exactly one of them occurs is given by:
\[ P(A \cap B') + P(A' \cap B = 26/49 \]
This can be expressed as:
\[ ab' + a'b = a(1-b) + b(1-a) = 26/49 \]
Simplifying this gives:
\[ a + b - 2ab = \frac{26}{49} \quad \text{(Equation 1)} \]

3. **Probability that None Occurs:**
The probability that none of them occurs is given by:
\[ P(A') \cap P(B') = (1 - a)(1 - b) = 15/49 \]
Expanding this gives:
\[ 1 - a - b + ab = \frac{15}{49} \]
Rearranging this gives:
\[ -a - b + ab = \frac{15}{49} - 1 \]
Simplifying this leads to:
\[ ab - a - b = -\frac{34}{49} \quad \text{(Equation 2)} \]

4. **Combine Equations:**
Now we have two equations:
- From Equation 1: \( a + b - 2ab = \frac{26}{49} \)
- From Equation 2: \( ab - a - b = -\frac{34}{49} \)

We can rewrite Equation 2 as:
\[ ab - a - b = -\frac{34}{49} \implies ab - a - b + 34/49 = 0 \]

5. **Substituting and Solving:**
From Equation 1, we can express \( a + b \):
\[ a + b = 2ab + \frac{26}{49} \]
Substitute this into Equation 2:
\[ ab - (2ab + \frac{26}{49}) = -\frac{34}{49} \]
Simplifying gives:
\[ -ab - \frac{26}{49} = -\frac{34}{49} \implies ab = \frac{8}{49} \]

6. **Finding \( a + b \):**
Substitute \( ab = \frac{8}{49} \) back into Equation 1:
\[ a + b = 2 \cdot \frac{8}{49} + \frac{26}{49} = \frac{16}{49} + \frac{26}{49} = \frac{42}{49} \]

7. **Using the Quadratic Formula:**
We have:
- \( a + b = \frac{42}{49} \)
- \( ab = \frac{8}{49} \)

The values of \( a \) and \( b \) are the roots of the quadratic equation:
\[ x^2 - (a+b)x + ab = 0 \implies x^2 - \frac{42}{49}x + \frac{8}{49} = 0 \]

8. **Calculating the Roots:**
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{\frac{42}{49} \pm \sqrt{\left(\frac{42}{49}\right)^2 - 4 \cdot \frac{8}{49}}}{2} \]
Simplifying gives:
\[ x = \frac{\frac{42}{49} \pm \sqrt{\frac{1764}{2401} - \frac{32}{49}}}{2} \]
Continuing to simplify leads to:
\[ x = \frac{\frac{42}{49} \pm \frac{14}{49}}{2} \]

9. **Finding the Probabilities:**
This results in:
- \( a = \frac{4}{7} \)
- \( b = \frac{2}{7} \)

10. **Conclusion:**
The probability of the more probable event is:
\[ \text{The answer is } \frac{4}{7}. \]