If `P (A) = (3)/(10) , P (B) = (2)/(5) and P ( A uuB) = (3)/(5),` then `P ((B)/(A)) + P ((A)/(B))` is equal to

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:

- \( P(A) = \frac{3}{10} \)
- \( P(B) = \frac{2}{5} \)
- \( P(A \cup B) = \frac{3}{5} \)

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

Using the formula for the probability of the union of two events:

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

Substituting the given values:

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

### Step 2: Convert \( P(B) \) to a common denominator

To perform the addition, we convert \( P(B) \):

\[ P(B) = \frac{2}{5} = \frac{4}{10} \]

### Step 3: Substitute and solve for \( P(A \cap B) \)

Now substituting back into the equation:

\[ \frac{3}{5} = \frac{3}{10} + \frac{4}{10} - P(A \cap B) \]

Combine the fractions on the right side:

\[ \frac{3}{5} = \frac{7}{10} - P(A \cap B) \]

### Step 4: Isolate \( P(A \cap B) \)

Now, we can isolate \( P(A \cap B) \):

\[ P(A \cap B) = \frac{7}{10} - \frac{3}{5} \]

Convert \( \frac{3}{5} \) to have a common denominator of 10:

\[ \frac{3}{5} = \frac{6}{10} \]

So,

\[ P(A \cap B) = \frac{7}{10} - \frac{6}{10} = \frac{1}{10} \]

### Step 5: Calculate \( P(B|A) \)

Using the formula for conditional probability:

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

Substituting the values we have:

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

### Step 6: Calculate \( P(A|B) \)

Similarly, we find \( P(A|B) \):

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

Substituting the values:

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

### Step 7: Add \( P(B|A) \) and \( P(A|B) \)

Now we can add the two conditional probabilities:

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

### Step 8: Find a common denominator and add

The common denominator for 3 and 4 is 12:

\[ \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12} \]

Thus,

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

### Final Answer

So, \( P(B|A) + P(A|B) = \frac{7}{12} \).

---