Question

If \( P(B) = \frac{3}{5} \), \( P(A \mid B) = \frac{1}{2} \), and \( P(A \cup B) = \frac{4}{5} \), then the value of \( P(A \cup B)' + P(A' \cup B) \) is:

• A. \( \frac{1}{5} \)
• B. \( \frac{4}{5} \)
• C. \( \frac{1}{2} \)
• D. \( 1 \)

Hint:

For probability problems involving unions and complements: 
- Use \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). 
- The complement rule: \( P(A^c) = 1 - P(A) \). - For conditional probability: \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \).

Answer 1

The Correct Option is D

Step 1: Compute \( P(A \cap B) \). Using the conditional probability formula: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \] Substituting the given values: \[ \frac{1}{2} = \frac{P(A \cap B)}{3/5} \] \[ P(A \cap B) = \frac{1}{2} \times \frac{3}{5} = \frac{3}{10}. \] Step 2: Compute \( P(A) \). Using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substituting known values: \[ \frac{4}{5} = P(A) + \frac{3}{5} - \frac{3}{10} \] \[ P(A) = \frac{4}{5} - \frac{3}{5} + \frac{3}{10} \] \[ P(A) = \frac{1}{5} + \frac{3}{10} = \frac{2}{10} + \frac{3}{10} = \frac{5}{10} = \frac{1}{2}. \] Step 3: Compute \( P(A \cup B)' \). \[ P(A \cup B)' = 1 - P(A \cup B) = 1 - \frac{4}{5} = \frac{1}{5}. \] Step 4: Compute \( P(A' \cup B) \). Using: \[ P(A' \cup B) = 1 - P(A \cap B'). \] First, compute \( P(A \cap B') \): \[ P(A \cap B') = P(A) - P(A \cap B) = \frac{1}{2} - \frac{3}{10} = \frac{5}{10} - \frac{3}{10} = \frac{2}{10} = \frac{1}{5}. \] Now: \[ P(A' \cup B) = 1 - \frac{1}{5} = \frac{4}{5}. \] Step 5: Compute \( P(A \cup B)' + P(A' \cup B) \). \[ P(A \cup B)' + P(A' \cup B) = \frac{1}{5} + \frac{4}{5} = 1. \] Thus, the correct answer is: \[ 1. \]