If A and B are two independent events such that `P(A) gt (1)/(2), P(A nn B^(C ))=(3)/(25) and P(A^(C )nnB)=(8)/(25)`, then `P(A)` is equal to `("where, "A^("c") and B^("c")` represent the complement of events A and B respectively)

Options

A. `(1)/(5)`
B. `(3)/(5)`
C. `(3)/(4)`
D. `(4)/(5)`

Answer

To solve the problem, we need to find the probability of event A given the conditions provided. Let's break down the solution step by step.

### Step 1: Understand the Given Information
We are given:
- \( P(A) > \frac{1}{2} \)
- \( P(A \cap B^C) = \frac{3}{25} \)
- \( P(A^C \cap B) = \frac{8}{25} \)

### Step 2: Use the Independence of Events
Since A and B are independent events, we can express the probabilities of their intersections:
1. \( P(A \cap B^C) = P(A) \cdot P(B^C) = P(A) \cdot (1 - P(B)) \)
2. \( P(A^C \cap B) = P(A^C) \cdot P(B) = (1 - P(A)) \cdot P(B) \)

### Step 3: Set Up the Equations
From the information given, we can set up two equations:
1. \( P(A) \cdot (1 - P(B)) = \frac{3}{25} \) (Equation 1)
2. \( (1 - P(A)) \cdot P(B) = \frac{8}{25} \) (Equation 2)

### Step 4: Express \( P(B) \) in Terms of \( P(A) \)
From Equation 1:
\[ P(A) - P(A) \cdot P(B) = \frac{3}{25} \]
Rearranging gives:
\[ P(A) \cdot P(B) = P(A) - \frac{3}{25} \tag{3} \]

From Equation 2:
\[ P(B) - P(A) \cdot P(B) = \frac{8}{25} \]
Rearranging gives:
\[ P(B) \cdot (1 - P(A)) = \frac{8}{25} \tag{4} \]

### Step 5: Substitute and Solve
From Equation (3), substitute \( P(B) \) from Equation (4):
Let \( P(B) = t \):
\[ t \cdot (1 - P(A)) = \frac{8}{25} \]
Substituting \( P(B) \) from Equation (3) into this gives:
\[ t - P(A) \cdot t = \frac{8}{25} \]

### Step 6: Solve for \( P(A) \) and \( P(B) \)
Now we have two equations:
1. \( P(A) \cdot (1 - t) = \frac{3}{25} \)
2. \( t \cdot (1 - P(A)) = \frac{8}{25} \)

By manipulating these equations, we can find values for \( P(A) \) and \( P(B) \).

### Step 7: Find Values
Let \( P(B) = t \):
From \( t - P(A) \cdot t = \frac{8}{25} \):
\[ t = P(A) + \frac{1}{5} \]

Substituting back into the equations will yield two possible values for \( P(A) \):
1. \( P(A) = \frac{3}{5} \)
2. \( P(A) = \frac{1}{5} \)

### Step 8: Choose the Correct Value
Since we know \( P(A) > \frac{1}{2} \), we select:
\[ P(A) = \frac{3}{5} \]

### Final Answer
Thus, the probability \( P(A) \) is:
\[ \boxed{\frac{3}{5}} \]