If `P(A)=(1)/(4),P(B)=(1)/(2)andP(AcapB)=(1)/(8)`, find (a) `P(AuuB).

Answer

To find \( P(A \cup B) \), we can use the formula:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Given:
- \( P(A) = \frac{1}{4} \)
- \( P(B) = \frac{1}{2} \)
- \( P(A \cap B) = \frac{1}{8} \)

Now, we will substitute these values into the formula.

### Step 1: Substitute the values into the formula

\[ P(A \cup B) = \frac{1}{4} + \frac{1}{2} - \frac{1}{8} \]

### Step 2: Find a common denominator

The common denominator for \( 4, 2, \) and \( 8 \) is \( 8 \).

### Step 3: Convert each fraction to have the common denominator

- Convert \( \frac{1}{4} \) to eighths:
\[ \frac{1}{4} = \frac{2}{8} \]
- Convert \( \frac{1}{2} \) to eighths:
\[ \frac{1}{2} = \frac{4}{8} \]
- \( \frac{1}{8} \) is already in eighths.

### Step 4: Substitute back into the equation

Now we substitute these values back into the equation:

\[ P(A \cup B) = \frac{2}{8} + \frac{4}{8} - \frac{1}{8} \]

### Step 5: Perform the addition and subtraction

Combine the fractions:

\[ P(A \cup B) = \frac{2 + 4 - 1}{8} = \frac{5}{8} \]

### Final Answer

Thus, the probability \( P(A \cup B) \) is:

\[ \boxed{\frac{5}{8}} \]