If A and B are two events such that `P(A)=(4)/(7), P(AnnB)=(3)/(28)` and the conditional probability `P((A)/(A^(c )uuB^(c )))` (where `A^(c )` denotes the compliment of the event A) is equal to `lambda`, then the value of `(26)/(lambda)` is equal to

Answer

To solve the problem step by step, we need to find the conditional probability \( P\left(A \mid (A^c \cup B^c)\right) \) and then determine the value of \( \frac{26}{\lambda} \).

### Step 1: Understand the Given Information
We have:
- \( P(A) = \frac{4}{7} \)
- \( P(A \cap B) = \frac{3}{28} \)

We need to find \( P\left(A \mid (A^c \cup B^c)\right) \).

### Step 2: Use the Definition of Conditional Probability
The conditional probability can be expressed as:
\[ P(A \mid (A^c \cup B^c)) = \frac{P(A \cap (A^c \cup B^c))}{P(A^c \cup B^c)} \]

### Step 3: Simplify \( A \cap (A^c \cup B^c) \)
Using the distributive property of sets, we have:
\[ A \cap (A^c \cup B^c) = (A \cap A^c) \cup (A \cap B^c) = \emptyset \cup (A \cap B^c) = A \cap B^c \]
So, we can rewrite the conditional probability as:
\[ P(A \mid (A^c \cup B^c)) = \frac{P(A \cap B^c)}{P(A^c \cup B^c)} \]

### Step 4: Find \( P(A \cap B^c) \)
Using the formula for the complement:
\[ P(A \cap B^c) = P(A) - P(A \cap B) \]
Substituting the known values:
\[ P(A \cap B^c) = \frac{4}{7} - \frac{3}{28} \]

To perform this subtraction, we need a common denominator. The least common multiple of 7 and 28 is 28:
\[ P(A \cap B^c) = \frac{16}{28} - \frac{3}{28} = \frac{13}{28} \]

### Step 5: Find \( P(A^c \cup B^c) \)
Using the formula for the union of complements:
\[ P(A^c \cup B^c) = 1 - P(A \cap B) \]
Substituting the known value:
\[ P(A^c \cup B^c) = 1 - \frac{3}{28} = \frac{28 - 3}{28} = \frac{25}{28} \]

### Step 6: Substitute Back into the Conditional Probability
Now we can substitute back into our expression for the conditional probability:
\[ P(A \mid (A^c \cup B^c)) = \frac{P(A \cap B^c)}{P(A^c \cup B^c)} = \frac{\frac{13}{28}}{\frac{25}{28}} = \frac{13}{25} \]

### Step 7: Set \( \lambda \) Equal to the Result
From the problem, we have \( \lambda = \frac{13}{25} \).

### Step 8: Calculate \( \frac{26}{\lambda} \)
Now we need to find:
\[ \frac{26}{\lambda} = \frac{26}{\frac{13}{25}} = 26 \times \frac{25}{13} = 2 \times 25 = 50 \]

### Final Answer
Thus, the value of \( \frac{26}{\lambda} \) is \( 50 \).