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 will use the given probabilities and apply the formula for conditional probability.

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

We need to find the conditional probability \( P(A | (A^c \cup B^c)) \) and set it equal to \( \lambda \).

### Step 2: Use the Formula for Conditional Probability
The formula for conditional probability is given by:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
In our case, we need to find:
\[ P(A | (A^c \cup B^c)) = \frac{P(A \cap (A^c \cup B^c))}{P(A^c \cup B^c)} \]

### Step 3: Simplify \( P(A \cap (A^c \cup B^c)) \)
Using the distributive property:
\[ P(A \cap (A^c \cup B^c)) = P((A \cap A^c) \cup (A \cap B^c)) \]
Since \( A \cap A^c \) is an empty set, we have:
\[ P(A \cap (A^c \cup B^c)) = P(A \cap B^c) \]

### Step 4: Calculate \( P(A \cap B^c) \)
Using the formula:
\[ P(A \cap B^c) = P(A) - P(A \cap B) \]
Substituting the values:
\[ P(A \cap B^c) = \frac{4}{7} - \frac{3}{28} \]
To subtract these fractions, we need a common denominator. The least common multiple of 7 and 28 is 28. Thus:
\[ P(A \cap B^c) = \frac{4 \times 4}{28} - \frac{3}{28} = \frac{16}{28} - \frac{3}{28} = \frac{13}{28} \]

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

### Step 6: Substitute into the Conditional Probability Formula
Now we can substitute back into the conditional probability formula:
\[ P(A | (A^c \cup B^c)) = \frac{P(A \cap B^c)}{P(A^c \cup B^c)} = \frac{\frac{13}{28}}{\frac{25}{28}} = \frac{13}{25} \]
Thus, \( \lambda = \frac{13}{25} \).

### Step 7: Find \( \frac{26}{\lambda} \)
Now we need to find:
\[ \frac{26}{\lambda} = \frac{26}{\frac{13}{25}} = 26 \times \frac{25}{13} = 2 \times 25 = 50 \]

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

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