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

We are given the following:
\( P(E) = 0.3 \)
\( P(E \cup F) = 0.5 \)
\( E \) and \( F \) are independent events.
We are asked to find \( P(E \mid F) - P(F \mid E) \).

Step 1: Use the formula for the union of two events.
We know the formula for the union of two events: \[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \] Since \( E \) and \( F \) are independent, we know that: \[ P(E \cap F) = P(E) \cdot P(F) \] Substituting the values: \[ 0.5 = 0.3 + P(F) - (0.3 \cdot P(F)) \] Simplifying: \[ 0.5 = 0.3 + P(F)(1 - 0.3) \] \[ 0.5 = 0.3 + 0.7 P(F) \] \[ 0.2 = 0.7 P(F) \] \[ P(F) = \frac{0.2}{0.7} = \frac{2}{7} \]
Step 2: Calculate \( P(E \mid F) \).
By the definition of conditional probability: \[ P(E \mid F) = \frac{P(E \cap F)}{P(F)} = \frac{P(E) \cdot P(F)}{P(F)} = P(E) = 0.3 \]
Step 3: Calculate \( P(F \mid E) \).
Similarly, by the definition of conditional probability: \[ P(F \mid E) = \frac{P(E \cap F)}{P(E)} = \frac{P(E) \cdot P(F)}{P(E)} = P(F) = \frac{2}{7} \]
Step 4: Find the difference.
Now we can find the required value: \[ P(E \mid F) - P(F \mid E) = 0.3 - \frac{2}{7} \] Convert 0.3 to a fraction: \[ P(E \mid F) - P(F \mid E) = \frac{3}{10} - \frac{2}{7} \] To subtract these fractions, find the common denominator (LCM of 10 and 7 is 70): \[ P(E \mid F) - P(F \mid E) = \frac{21}{70} - \frac{20}{70} = \frac{1}{70} \]
Final Answer: \( \boxed{\frac{1}{70}} \).