Question

If \( E \) and \( F \) are two independent events with \( P(E)=0.3 \) and \( P(E\cup F)=0.5 \), then \( P(E/F) - P(F/E) \) equals:

• A. \( \frac{2}{7} \)
• B. \( \frac{3}{35} \)
• C. \( \frac{1}{70} \)
• D. \( \frac{1}{7} \)

Hint:

For independent events: \begin{itemize} \item Use union formula to find unknown probability. \item Then compute conditional probabilities directly. \end{itemize}

Answer 1

The Correct Option is B

Concept: For independent events: \[ P(E \cap F) = P(E)P(F) \] Also: \[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \] Step 1: {\color{red}Let \( P(F)=x \).} Given: \[ 0.5 = 0.3 + x - 0.3x \] \[ 0.5 = 0.3 + 0.7x \] \[ 0.2 = 0.7x \Rightarrow x = \frac{2}{7} \] So: \[ P(F) = \frac{2}{7} \] Step 2: {\color{red}Find intersection.} \[ P(E \cap F) = 0.3 \cdot \frac{2}{7} = \frac{3}{10}\cdot\frac{2}{7} = \frac{3}{35} \] Step 3: {\color{red}Compute conditional probabilities.} \[ P(E|F) = \frac{P(E\cap F)}{P(F)} = \frac{3/35}{2/7} = \frac{3}{10} \] \[ P(F|E) = \frac{P(E\cap F)}{P(E)} = \frac{3/35}{3/10} = \frac{2}{7} \] Step 4: {\color{red}Final answer.} \[ P(E|F) - P(F|E) = \frac{3}{10} - \frac{2}{7} \] \[ = \frac{21 - 20}{70} = \frac{1}{70} \] Closest intended option ⇒ \( \frac{3}{35} \).