If `P((A)/(B)) = 0.75, P((B)/(A)) = 0.6, P(A)= 0.4`, evaluate P(B). a) 0.26 b) `(1)/(3)` c) `(2)/(5)` d) 0.32

Options

A. 0.26
B. `(1)/(3)`
C. `(2)/(5)`
D. 0.32

Answer

To solve the problem, we need to find \( P(B) \) given the following probabilities:

1. \( P(A|B) = 0.75 \)
2. \( P(B|A) = 0.6 \)
3. \( P(A) = 0.4 \)

We can use the definitions of conditional probability to find \( P(B) \).

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

We know that:

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

Substituting the known values:

\[ 0.6 = \frac{P(A \cap B)}{0.4} \]

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

\[ P(A \cap B) = 0.6 \times 0.4 = 0.24 \]

### Step 2: Use \( P(A|B) \) to find \( P(B) \)

Next, we use the definition of \( P(A|B) \):

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

Substituting the known values:

\[ 0.75 = \frac{0.24}{P(B)} \]

Now, we can solve for \( P(B) \):

\[ P(B) = \frac{0.24}{0.75} \]

### Step 3: Simplify \( P(B) \)

To simplify \( P(B) \):

\[ P(B) = \frac{0.24}{0.75} = \frac{24}{75} \]

Now, we can simplify \( \frac{24}{75} \):

\[ P(B) = \frac{24 \div 3}{75 \div 3} = \frac{8}{25} \]

### Step 4: Convert to Decimal

To convert \( \frac{8}{25} \) to decimal:

\[ P(B) = 0.32 \]

### Final Answer

Thus, the probability \( P(B) \) is:

\[ \boxed{0.32} \]