If `P(A cap B)=(1)/(2), P(overline(A) cap overline(B))=(1)/(2)` and 2P(A)=P(B)=p, then the value of p is equal to

Options

A. `(1)/(2)`
B. `(2)/(3)`
C. `(1)/(4)`
D. `(1)/(3)`

Explanation

To solve the problem, we need to find the value of \( p \) given the following conditions:

1. \( P(A \cap B) = \frac{1}{2} \)
2. \( P(\overline{A} \cap \overline{B}) = \frac{1}{2} \)
3. \( 2P(A) = P(B) = p \)

Let's denote:
- \( P(A) = \frac{p}{2} \)
- \( P(B) = p \)

### Step 1: Use the relationship between probabilities
From the second condition, we know that:
\[ P(\overline{A} \cap \overline{B}) = \frac{1}{2} \]
Using the property of probabilities, we can express this as:
\[ P(\overline{A} \cap \overline{B}) = 1 - P(A \cup B) \]
Thus, we have:
\[ 1 - P(A \cup B) = \frac{1}{2} \]
This implies:
\[ P(A \cup B) = \frac{1}{2} \]

### Step 2: Use the formula for \( P(A \cup B) \)
We know that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ \frac{1}{2} = P(A) + P(B) - P(A \cap B) \]
Substituting \( P(A) = \frac{p}{2} \), \( P(B) = p \), and \( P(A \cap B) = \frac{1}{2} \):
\[ \frac{1}{2} = \frac{p}{2} + p - \frac{1}{2} \]

### Step 3: Simplify the equation
Now, we can simplify the equation:
\[ \frac{1}{2} = \frac{p}{2} + p - \frac{1}{2} \]
Combining like terms:
\[ \frac{1}{2} + \frac{1}{2} = \frac{p}{2} + p \]
This simplifies to:
\[ 1 = \frac{p}{2} + p \]
Expressing \( p \) in terms of a single fraction:
\[ 1 = \frac{p + 2p}{2} = \frac{3p}{2} \]

### Step 4: Solve for \( p \)
Now, we can solve for \( p \):
\[ 3p = 2 \implies p = \frac{2}{3} \]

Thus, the value of \( p \) is \( \frac{2}{3} \).

### Final Answer
The value of \( p \) is \( \frac{2}{3} \).

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