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 problem step by step, we will use the information given about two independent events A and B.

### Step 1: Define the probabilities
Let \( P(A) = p \) and \( P(B) = q \). We know from the problem that:
1. The probability that both events occur is given by:
\[ P(A \cap B) = p \cdot q = \frac{8}{49} \quad \text{(Equation 1)} \]

### Step 2: Probability of exactly one event occurring
The probability that exactly one of the events occurs can be expressed as:
\[ P(A \cap B') + P(A' \cap B) = P(A) + P(B) - 2P(A \cap B) \]
This is given as:
\[ P(A \cap B') + P(A' \cap B) = \frac{26}{49} \quad \text{(Equation 2)} \]
Substituting Equation 1 into Equation 2, we get:
\[ p + q - 2 \cdot \frac{8}{49} = \frac{26}{49} \]
This simplifies to:
\[ p + q - \frac{16}{49} = \frac{26}{49} \]
\[ p + q = \frac{26}{49} + \frac{16}{49} = \frac{42}{49} = \frac{6}{7} \]

### Step 3: Set up the equations
Now we have two equations:
1. \( p \cdot q = \frac{8}{49} \)
2. \( p + q = \frac{6}{7} \)

### Step 4: Solve for p and q
Let’s denote \( p \) and \( q \) as the roots of the quadratic equation:
\[ x^2 - (p + q)x + pq = 0 \]
Substituting the values we found:
\[ x^2 - \frac{6}{7}x + \frac{8}{49} = 0 \]

### Step 5: Multiply through by 49 to eliminate the fraction
\[ 49x^2 - 42x + 8 = 0 \]

### Step 6: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{42 \pm \sqrt{(-42)^2 - 4 \cdot 49 \cdot 8}}{2 \cdot 49} \]
Calculating the discriminant:
\[ = 1764 - 1568 = 196 \]
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 7: Identify \( \lambda \)
Thus, we have:
- \( P(A) = \frac{4}{7} \) and \( P(B) = \frac{2}{7} \) or vice versa.
- The more probable event has probability \( \lambda = \frac{4}{7} \).

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

### Final Answer
Thus, the value of \( 14\lambda \) is \( 8 \).

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