Question

Given that A and B are two events such that P(B) = \(\frac{3}{5}\) P(A/B) = \(\frac{1}{2}\) and P(A ∪ B) = \(\frac{4}{5}\) then P(A) =

• A. \(\frac{3}{10}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{1}{5}\)
• D. \(\frac{3}{5}\)

Answer 1

The Correct Option is B

We are given the following information: \[ P(B) = \frac{3}{5}, \quad P(A/B) = \frac{1}{2}, \quad P(A \cup B) = \frac{4}{5}. \] From the conditional probability formula: \[ P(A/B) = \frac{P(A \cap B)}{P(B)}. \] Substituting the given values: \[ \frac{1}{2} = \frac{P(A \cap B)}{\frac{3}{5}}, \] Solving for \( P(A \cap B) \): \[ P(A \cap B) = \frac{1}{2} \times \frac{3}{5} = \frac{3}{10}. \] Next, we use the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \] Substituting the given values: \[ \frac{4}{5} = P(A) + \frac{3}{5} - \frac{3}{10}. \] Multiplying the entire equation by 10 to clear the denominators: \[ 8 = 10P(A) + 6 - 3. \] Simplifying: \[ 8 = 10P(A) + 3 \quad \Rightarrow \quad 10P(A) = 5 \quad \Rightarrow \quad P(A) = \frac{1}{2}. \]

So, the correct answer is (B) : \(\frac{1}{2}\).

Answer 2

Given: 

• \(P(B) = \frac{3}{5}\)
• \(P(A|B) = \frac{1}{2}\)
• \(P(A \cup B) = \frac{4}{5}\)

Step 1: Use the conditional probability formula:

\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)

\(\Rightarrow \frac{1}{2} = \frac{P(A \cap B)}{3/5}\)

\(\Rightarrow P(A \cap B) = \frac{1}{2} \cdot \frac{3}{5} = \frac{3}{10}\)

Step 2: Use the formula for union of two events:

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

\(\frac{4}{5} = P(A) + \frac{3}{5} - \frac{3}{10}\)

Simplify:

\(\frac{4}{5} = P(A) + \frac{6}{10} - \frac{3}{10} = P(A) + \frac{3}{10}\)

\(\Rightarrow P(A) = \frac{4}{5} - \frac{3}{10} = \frac{8}{10} - \frac{3}{10} = \frac{5}{10} = \frac{1}{2}\)

Answer: \(\frac{1}{2}\)