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

Options

A. `1/4`
B. `1/3`
C. `5/12`
D. `7/12`

Answer

To solve the problem, we need to find \( P(B|A) + P(A|B) \) given the probabilities \( P(A) = \frac{3}{10} \), \( P(B) = \frac{2}{5} \), and \( P(A \cup B) = \frac{3}{5} \).

### Step 1: Find \( P(A \cap B) \)

We can use the formula for the probability of the intersection of two events:

\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]

Substituting the given values:

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

### Step 2: Convert fractions to a common denominator

The common denominator for \( 10 \) and \( 5 \) is \( 10 \). We convert \( \frac{2}{5} \) and \( \frac{3}{5} \):

\[ P(A \cap B) = \frac{3}{10} + \frac{4}{10} - \frac{6}{10} \]

### Step 3: Simplify the expression

Now, we can simplify:

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

### Step 4: Calculate \( P(B|A) \) and \( P(A|B) \)

Using the definition of conditional probability:

\[ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{1}{10}}{\frac{3}{10}} = \frac{1}{3} \]

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{10}}{\frac{2}{5}} = \frac{1}{10} \times \frac{5}{2} = \frac{1}{4} \]

### Step 5: Add \( P(B|A) \) and \( P(A|B) \)

Now we can add the two probabilities:

\[ P(B|A) + P(A|B) = \frac{1}{3} + \frac{1}{4} \]

### Step 6: Find a common denominator for the addition

The common denominator for \( 3 \) and \( 4 \) is \( 12 \):

\[ P(B|A) + P(A|B) = \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \]

### Final Answer

Thus, the final answer is:

\[ P(B|A) + P(A|B) = \frac{7}{12} \]

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