If `P(A)= (1)/(2), P((B)/(A)) = (1)/(3), P((A)/(B))= (2)/(5)`, find P(B)

Answer

To solve the problem, we need to find \( P(B) \) using the given probabilities. Let's break it down step by step.

### Step 1: Write down the given probabilities.
We have:
- \( P(A) = \frac{1}{2} \)
- \( P(B|A) = \frac{1}{3} \)
- \( P(A|B) = \frac{2}{5} \)

### Step 2: Use the definition of conditional probability.
The conditional probability \( P(B|A) \) can be expressed as:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]
From this, we can rearrange to find \( P(A \cap B) \):
\[ P(A \cap B) = P(B|A) \cdot P(A) \]

### Step 3: Substitute the known values.
Substituting the known values into the equation:
\[ P(A \cap B) = \frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6} \]

### Step 4: Use the definition of conditional probability for \( P(A|B) \).
The conditional probability \( P(A|B) \) can be expressed as:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Rearranging gives us:
\[ P(A \cap B) = P(A|B) \cdot P(B) \]

### Step 5: Substitute \( P(A \cap B) \) into the equation.
We already found \( P(A \cap B) = \frac{1}{6} \), so we can substitute this value:
\[ \frac{1}{6} = \frac{2}{5} \cdot P(B) \]

### Step 6: Solve for \( P(B) \).
To find \( P(B) \), we can rearrange the equation:
\[ P(B) = \frac{1}{6} \cdot \frac{5}{2} \]
Calculating this gives:
\[ P(B) = \frac{5}{12} \]

### Final Answer:
Thus, the probability \( P(B) \) is:
\[ \boxed{\frac{5}{12}} \]