If P(A) `= 3/10 ,P(B) =2/5 and P(A cup B) = 3/5` , then `P(B//A) +P(A//B)` =

Options

A. `0.24`
B. `0.96`
C. `0.3`
D. `0.58`

Answer

To solve the problem step by step, we will use the given probabilities and apply the formulas for conditional probability and the principle of inclusion-exclusion.

### Step 1: Identify the given probabilities
We are given:
- \( P(A) = \frac{3}{10} \)
- \( P(B) = \frac{2}{5} \)
- \( P(A \cup B) = \frac{3}{5} \)

### Step 2: Find \( P(A \cap B) \)
Using the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
We can rearrange this to find \( P(A \cap B) \):
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]

Substituting the values:
\[ P(A \cap B) = \frac{3}{10} + \frac{2}{5} - \frac{3}{5} \]

### Step 3: Convert fractions to a common denominator
The common denominator for \( 10 \) and \( 5 \) is \( 10 \). We convert \( \frac{2}{5} \) and \( \frac{3}{5} \) to have a denominator of \( 10 \):
\[ \frac{2}{5} = \frac{4}{10}, \quad \frac{3}{5} = \frac{6}{10} \]

Now substituting these values:
\[ P(A \cap B) = \frac{3}{10} + \frac{4}{10} - \frac{6}{10} \]
\[ P(A \cap B) = \frac{3 + 4 - 6}{10} = \frac{1}{10} \]

### Step 4: Calculate \( P(B|A) \)
Using the formula for conditional probability:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]
Substituting the values we found:
\[ P(B|A) = \frac{\frac{1}{10}}{\frac{3}{10}} = \frac{1}{3} \]

### Step 5: Calculate \( P(A|B) \)
Similarly, we calculate \( P(A|B) \):
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Substituting the values:
\[ P(A|B) = \frac{\frac{1}{10}}{\frac{2}{5}} = \frac{1}{10} \times \frac{5}{2} = \frac{1}{4} \]

### Step 6: Find \( P(B|A) + P(A|B) \)
Now, we add the two conditional probabilities:
\[ P(B|A) + P(A|B) = \frac{1}{3} + \frac{1}{4} \]

### Step 7: Find a common denominator and add
The common denominator for \( 3 \) and \( 4 \) is \( 12 \):
\[ \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12} \]
Now adding:
\[ P(B|A) + P(A|B) = \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \]

### Final Answer
Thus, the final answer is:
\[ P(B|A) + P(A|B) = \frac{7}{12} \]