Question

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

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

Hint:

For problems involving independent events, remember that \( P(E \cap F) = P(E) \cdot P(F) \). Always simplify the conditional probabilities carefully, especially when calculating differences.

Answer 1

The Correct Option is C

Given the following:
\( P(E) = 0.3 \)
\( P(E \cup F) = 0.5 \)
\( E \) and \( F \) are independent.

Find \( P(E \mid F) - P(F \mid E) \).

Step 1: Calculate \(P(F)\).
Use the union formula:\ \[\nP(E \cup F) = P(E) + P(F) - P(E \cap F)\n\] Since \( E \) and \( F \) are independent:\ \[\nP(E \cap F) = P(E) \cdot P(F)\n\] Substitute values:\ \[\n0.5 = 0.3 + P(F) - (0.3 \cdot P(F))\n\] Simplify:\ \[\n0.5 = 0.3 + P(F)(1 - 0.3)\n\] \[\n0.5 = 0.3 + 0.7 P(F)\n\] \[\n0.2 = 0.7 P(F)\n\] \[\nP(F) = \frac{0.2}{0.7} = \frac{2}{7}\n\]
Step 2: Calculate \( P(E \mid F) \).
By definition:\ \[\nP(E \mid F) = \frac{P(E \cap F)}{P(F)} = \frac{P(E) \cdot P(F)}{P(F)} = P(E) = 0.3\n\]
Step 3: Calculate \( P(F \mid E) \).
By definition:\ \[\nP(F \mid E) = \frac{P(E \cap F)}{P(E)} = \frac{P(E) \cdot P(F)}{P(E)} = P(F) = \frac{2}{7}\n\]
Step 4: Find the difference.
Calculate the required value:\ \[\nP(E \mid F) - P(F \mid E) = 0.3 - \frac{2}{7}\n\] Convert to fractions:\ \[\nP(E \mid F) - P(F \mid E) = \frac{3}{10} - \frac{2}{7}\n\] Common denominator (70):\ \[\nP(E \mid F) - P(F \mid E) = \frac{21}{70} - \frac{20}{70} = \frac{1}{70}\n\]
Final Answer: \( \boxed{\frac{1}{70}} \).