A and B are two non mutually exclusive events. If `P(A)=1/4, P(B)=2/5`, and `P(AuuB)=1/2`, find the values of `P(AnnB)` and `P(AnnbarB)`

Answer

To solve the problem, we need to find the values of \( P(A \cap B) \) and \( P(A \cap B') \) using the given probabilities.

### Given:
- \( P(A) = \frac{1}{4} \)
- \( P(B) = \frac{2}{5} \)
- \( P(A \cup B) = \frac{1}{2} \)

### Step 1: Use the formula for the probability of the union of two events.

The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 2: Substitute the known values into the formula.

Substituting the values we have:
\[ \frac{1}{2} = \frac{1}{4} + \frac{2}{5} - P(A \cap B) \]

### Step 3: Convert all fractions to a common denominator.

The least common multiple of the denominators \( 2, 4, \) and \( 5 \) is \( 20 \). We convert each term:
- \( \frac{1}{2} = \frac{10}{20} \)
- \( \frac{1}{4} = \frac{5}{20} \)
- \( \frac{2}{5} = \frac{8}{20} \)

Now, substituting these into the equation gives:
\[ \frac{10}{20} = \frac{5}{20} + \frac{8}{20} - P(A \cap B) \]

### Step 4: Simplify the equation.

Combine the fractions on the right:
\[ \frac{10}{20} = \frac{13}{20} - P(A \cap B) \]

### Step 5: Isolate \( P(A \cap B) \).

Rearranging the equation gives:
\[ P(A \cap B) = \frac{13}{20} - \frac{10}{20} = \frac{3}{20} \]

### Step 6: Find \( P(A \cap B') \).

To find \( P(A \cap B') \), we use the formula:
\[ P(A \cap B') = P(A) - P(A \cap B) \]

### Step 7: Substitute the known values.

Substituting the known values:
\[ P(A \cap B') = \frac{1}{4} - P(A \cap B) = \frac{1}{4} - \frac{3}{20} \]

### Step 8: Convert \( \frac{1}{4} \) to a fraction with a denominator of \( 20 \).

\[ \frac{1}{4} = \frac{5}{20} \]

Now substituting gives:
\[ P(A \cap B') = \frac{5}{20} - \frac{3}{20} = \frac{2}{20} = \frac{1}{10} \]

### Final Answers:
- \( P(A \cap B) = \frac{3}{20} \)
- \( P(A \cap B') = \frac{1}{10} \)