If A and B are two events such that `P(A) = (3)/(8)` ` P(A) = (5)/(8)` and ` P(A uu B) = (3)/(4)` , then `P((bar(A))/(B))`

Options

A. `(2)/(5)`
B. `(3)/(5)`
C. `(3)/(20)`
D. `(6)/(25)`

Answer

To solve the problem step by step, we need to find \( P(\bar{A} | B) \) given the probabilities \( P(A) = \frac{3}{8} \), \( P(B) = \frac{5}{8} \), and \( P(A \cup B) = \frac{3}{4} \).

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

We use 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 known values:

\[ \frac{3}{4} = \frac{3}{8} + \frac{5}{8} - P(A \cap B) \]

### Step 2: Simplify the equation

Combine \( P(A) \) and \( P(B) \):

\[ \frac{3}{4} = \frac{3 + 5}{8} - P(A \cap B) \]

This simplifies to:

\[ \frac{3}{4} = \frac{8}{8} - P(A \cap B) \]

### Step 3: Solve for \( P(A \cap B) \)

Rearranging gives:

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

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

Using the definition of conditional probability:

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

Substituting the values we have:

\[ P(A | B) = \frac{\frac{1}{4}}{\frac{5}{8}} = \frac{1}{4} \times \frac{8}{5} = \frac{2}{5} \]

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

Using the relationship between \( P(A | B) \) and \( P(\bar{A} | B) \):

\[ P(\bar{A} | B) = 1 - P(A | B) \]

Substituting the value we found:

\[ P(\bar{A} | B) = 1 - \frac{2}{5} = \frac{3}{5} \]

### Final Step: Conclusion

Thus, the final answer is:

\[ P(\bar{A} | B) = \frac{3}{5} \]