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)/(38)`
C. `(1)/(70)`
D. `(1)/(7)`

Answer

To solve the problem step by step, we will use the given probabilities and the properties of independent events.

### Step 1: Identify Given Values
We are given:
- \( P(E) = 0.3 \)
- \( P(E \cup F) = 0.5 \)

### Step 2: Use the Formula for Union of Two Events
For any two events \( E \) and \( F \), the probability of their union is given by:
\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]
Since \( E \) and \( F \) are independent, we can express \( P(E \cap F) \) as:
\[ P(E \cap F) = P(E) \cdot P(F) \]

### Step 3: Substitute Values into the Union Formula
Substituting the known values into the union formula:
\[ 0.5 = P(E) + P(F) - P(E) \cdot P(F) \]
Let \( P(F) = x \). Then we have:
\[ 0.5 = 0.3 + x - 0.3x \]

### Step 4: Rearrange the Equation
Rearranging the equation gives:
\[ 0.5 = 0.3 + x - 0.3x \]
\[ 0.5 - 0.3 = x - 0.3x \]
\[ 0.2 = x(1 - 0.3) \]
\[ 0.2 = 0.7x \]

### Step 5: Solve for \( P(F) \)
Now, solve for \( x \):
\[ x = \frac{0.2}{0.7} = \frac{2}{7} \]
Thus, \( P(F) = \frac{2}{7} \).

### Step 6: Calculate \( P(E | F) \) and \( P(F | E) \)
Now we can calculate \( P(E | F) \) and \( P(F | E) \):
\[ P(E | F) = \frac{P(E \cap F)}{P(F)} = \frac{P(E) \cdot P(F)}{P(F)} = P(E) = 0.3 \]
\[ P(F | E) = \frac{P(F \cap E)}{P(E)} = \frac{P(E) \cdot P(F)}{P(E)} = P(F) = \frac{2}{7} \]

### Step 7: Calculate \( P(E | F) - P(F | E) \)
Now we find:
\[ P(E | F) - P(F | E) = 0.3 - \frac{2}{7} \]
To perform this subtraction, convert \( 0.3 \) into a fraction:
\[ 0.3 = \frac{3}{10} \]
Now we need a common denominator to subtract:
\[ P(E | F) - P(F | E) = \frac{3}{10} - \frac{2}{7} \]
The least common multiple of 10 and 7 is 70. Convert both fractions:
\[ \frac{3}{10} = \frac{21}{70}, \quad \frac{2}{7} = \frac{20}{70} \]
Thus,
\[ P(E | F) - P(F | E) = \frac{21}{70} - \frac{20}{70} = \frac{1}{70} \]

### Final Answer
The final answer is:
\[ P(E | F) - P(F | E) = \frac{1}{70} \]