Let A and B are two events such that P(exactly one) = `2/5 , P(A uu B)= 1/2` then `P(A nn B)=`

Options

A. `1/10`
B. `2/9`
C. `1/8`
D. `1/12`

Explanation

To solve the problem, we need to find \( P(A \cap B) \) given the following information:

1. \( P(\text{exactly one of A or B}) = \frac{2}{5} \)
2. \( P(A \cup B) = \frac{1}{2} \)

### Step-by-Step Solution:

**Step 1: Understand the formula for exactly one of A or B.**

The probability of exactly one of the events A or B occurring can be expressed as:
\[ P(\text{exactly one of A or B}) = P(A) + P(B) - 2P(A \cap B) \]
Given that this probability is \( \frac{2}{5} \), we can write:
\[ P(A) + P(B) - 2P(A \cap B) = \frac{2}{5} \quad \text{(Equation 1)} \]

**Step 2: Understand the formula for the union of A and B.**

The probability of the union of two events A and B can be expressed as:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Given that this probability is \( \frac{1}{2} \), we can write:
\[ P(A) + P(B) - P(A \cap B) = \frac{1}{2} \quad \text{(Equation 2)} \]

**Step 3: Set up the equations.**

From Equation 1:
\[ P(A) + P(B) = \frac{2}{5} + 2P(A \cap B) \quad \text{(Rearranging Equation 1)} \]

From Equation 2:
\[ P(A) + P(B) = \frac{1}{2} + P(A \cap B) \quad \text{(Rearranging Equation 2)} \]

**Step 4: Equate the two expressions for \( P(A) + P(B) \).**

Setting the two expressions for \( P(A) + P(B) \) equal to each other:
\[ \frac{2}{5} + 2P(A \cap B) = \frac{1}{2} + P(A \cap B) \]

**Step 5: Solve for \( P(A \cap B) \).**

Subtract \( P(A \cap B) \) from both sides:
\[ \frac{2}{5} + P(A \cap B) = \frac{1}{2} \]

Now, isolate \( P(A \cap B) \):
\[ P(A \cap B) = \frac{1}{2} - \frac{2}{5} \]

To perform the subtraction, convert \( \frac{1}{2} \) to a fraction with a denominator of 10:
\[ \frac{1}{2} = \frac{5}{10}, \quad \frac{2}{5} = \frac{4}{10} \]
Thus,
\[ P(A \cap B) = \frac{5}{10} - \frac{4}{10} = \frac{1}{10} \]

### Final Answer:
\[ P(A \cap B) = \frac{1}{10} \]