Two events E and F are independent. If `P(E )=0.3` and `P(EcupF)=0.5` then `P(E//F)-P(F//E)` equals to

Options

A. `2/7`
B. `3/35`
C. `1/70`
D. `1/7`

Explanation

To solve the problem, we need to find the value of \( P(E|F) - P(F|E) \) given that events \( E \) and \( F \) are independent, \( P(E) = 0.3 \), and \( P(E \cup F) = 0.5 \).

### Step-by-Step Solution:

1. **Understand the relationship between probabilities:**
Since \( E \) and \( F \) are independent events, we have:
\[ P(E \cap F) = P(E) \cdot P(F) \]

2. **Use the formula for the union of two events:**
The formula for the probability of the union of two events is:
\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]
Plugging in the known values:
\[ 0.5 = P(E) + P(F) - P(E \cap F) \]
Substituting \( P(E) = 0.3 \) and \( P(E \cap F) = P(E) \cdot P(F) = 0.3 \cdot P(F) \):
\[ 0.5 = 0.3 + P(F) - 0.3 \cdot P(F) \]

3. **Let \( P(F) = x \):**
We can rewrite the equation as:
\[ 0.5 = 0.3 + x - 0.3x \]
Simplifying this, we get:
\[ 0.5 = 0.3 + x(1 - 0.3) \]
\[ 0.5 = 0.3 + 0.7x \]

4. **Solve for \( x \):**
Rearranging gives:
\[ 0.5 - 0.3 = 0.7x \]
\[ 0.2 = 0.7x \]
\[ x = \frac{0.2}{0.7} = \frac{2}{7} \]
Thus, \( P(F) = \frac{2}{7} \).

5. **Calculate \( P(E \cap F) \):**
Now, we can find \( P(E \cap F) \):
\[ P(E \cap F) = P(E) \cdot P(F) = 0.3 \cdot \frac{2}{7} = \frac{0.6}{7} = \frac{6}{70} \]

6. **Calculate \( P(E|F) \) and \( P(F|E) \):**
Using the definition of conditional probability:
\[ P(E|F) = \frac{P(E \cap F)}{P(F)} = \frac{\frac{6}{70}}{\frac{2}{7}} = \frac{6}{70} \cdot \frac{7}{2} = \frac{6 \cdot 7}{70 \cdot 2} = \frac{21}{70} = \frac{3}{10} \]
\[ P(F|E) = \frac{P(E \cap F)}{P(E)} = \frac{\frac{6}{70}}{0.3} = \frac{\frac{6}{70}}{\frac{3}{10}} = \frac{6}{70} \cdot \frac{10}{3} = \frac{60}{210} = \frac{2}{7} \]

7. **Calculate \( P(E|F) - P(F|E) \):**
Now, we find the difference:
\[ P(E|F) - P(F|E) = \frac{3}{10} - \frac{2}{7} \]
To subtract these fractions, find a common denominator (which is 70):
\[ = \frac{3 \cdot 7}{10 \cdot 7} - \frac{2 \cdot 10}{7 \cdot 10} = \frac{21}{70} - \frac{20}{70} = \frac{1}{70} \]

### Final Answer:
Thus, \( P(E|F) - P(F|E) = \frac{1}{70} \).