Given two independent events, if the probability that both the events occur is `(8)/(49)`, the probability that exactly one of them occurs is `(26)/(49)` and the probability of more probable of the two events is `lambda`, then `14 lambda` is equal to

Options

A. 2
B. 4
C. 8
D. 7

Answer

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

### Step 1: Define the Events
Let \( A \) and \( B \) be two independent events. We know the following:
- The probability that both events occur, \( P(A \cap B) = P(A) \cdot P(B) = \frac{8}{49} \).
- The probability that exactly one of the events occurs is given by \( P(A \cup B) - P(A \cap B) = \frac{26}{49} \).

### Step 2: Set Up the Equations
From the information provided, we can set up the following equations:
1. \( P(A) \cdot P(B) = \frac{8}{49} \) (Equation 1)
2. The probability that exactly one of the events occurs can be expressed as:
\[ P(A) + P(B) - 2 \cdot P(A) \cdot P(B) = \frac{26}{49} \]
Substituting Equation 1 into this gives:
\[ P(A) + P(B) - 2 \cdot \frac{8}{49} = \frac{26}{49} \]

### Step 3: Simplify the Second Equation
Rearranging the second equation:
\[ P(A) + P(B) - \frac{16}{49} = \frac{26}{49} \]
Adding \( \frac{16}{49} \) to both sides:
\[ P(A) + P(B) = \frac{26}{49} + \frac{16}{49} = \frac{42}{49} = \frac{6}{7} \]
This gives us a second equation (Equation 2):
\[ P(A) + P(B) = \frac{6}{7} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( P(A) \cdot P(B) = \frac{8}{49} \)
2. \( P(A) + P(B) = \frac{6}{7} \)

Let \( P(A) = x \) and \( P(B) = y \). Then:
1. \( xy = \frac{8}{49} \)
2. \( x + y = \frac{6}{7} \)

From the second equation, we can express \( y \) in terms of \( x \):
\[ y = \frac{6}{7} - x \]
Substituting this into the first equation:
\[ x\left(\frac{6}{7} - x\right) = \frac{8}{49} \]
Expanding this gives:
\[ \frac{6x}{7} - x^2 = \frac{8}{49} \]
Multiplying through by \( 49 \) to eliminate the fraction:
\[ 49 \cdot \frac{6x}{7} - 49x^2 = 8 \]
This simplifies to:
\[ 42x - 49x^2 = 8 \]
Rearranging gives:
\[ 49x^2 - 42x + 8 = 0 \]

### Step 5: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 49 \), \( b = -42 \), and \( c = 8 \):
\[ x = \frac{42 \pm \sqrt{(-42)^2 - 4 \cdot 49 \cdot 8}}{2 \cdot 49} \]
Calculating the discriminant:
\[ (-42)^2 - 4 \cdot 49 \cdot 8 = 1764 - 1568 = 196 \]
Thus,
\[ x = \frac{42 \pm 14}{98} \]
Calculating the two possible values:
1. \( x = \frac{56}{98} = \frac{4}{7} \)
2. \( x = \frac{28}{98} = \frac{2}{7} \)

### Step 6: Determine the Probabilities
Thus, we have:
- If \( P(A) = \frac{4}{7} \), then \( P(B) = \frac{2}{7} \).
- If \( P(A) = \frac{2}{7} \), then \( P(B) = \frac{4}{7} \).

### Step 7: Find \( \lambda \)
The more probable of the two events is:
\[ \lambda = \frac{4}{7} \]

### Step 8: Calculate \( 14\lambda \)
Now, we need to find \( 14\lambda \):
\[ 14\lambda = 14 \cdot \frac{4}{7} = 8 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{8} \]