`P(A)=3//8,P(B)=1//2,P(AuuB)=5//8,` which of the following do/does hold good?

Options

A. `P(A^(C )//B)=2P(A//B^(C ))`
B. `P(B)=P(A//B)`
C. `15P(A^(c )//B^(C ))=8P(B//A^(C ))`
D. `P(A//B^(C ))=(AnnB)`

Explanation

To solve the problem step by step, we will use the given probabilities and apply the relevant probability formulas.

### Given:
- \( P(A) = \frac{3}{8} \)
- \( P(B) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{5}{8} \)

### Step 1: Find \( P(A \cap B) \)
We can use the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the given values:
\[ \frac{5}{8} = \frac{3}{8} + \frac{1}{2} - P(A \cap B) \]
Convert \( \frac{1}{2} \) to eighths:
\[ \frac{1}{2} = \frac{4}{8} \]
Now substitute:
\[ \frac{5}{8} = \frac{3}{8} + \frac{4}{8} - P(A \cap B) \]
Combine the fractions:
\[ \frac{5}{8} = \frac{7}{8} - P(A \cap B) \]
Now, isolate \( P(A \cap B) \):
\[ P(A \cap B) = \frac{7}{8} - \frac{5}{8} = \frac{2}{8} = \frac{1}{4} \]

### Step 2: Check Option 1
We need to check if:
\[ P(A' | B) = 2 \cdot P(A | B') \]
Where \( A' \) is the complement of A and \( B' \) is the complement of B.

#### Find \( P(A' | B) \):
Using the definition of conditional probability:
\[ P(A' | B) = \frac{P(A' \cap B)}{P(B)} = \frac{P(B) - P(A \cap B)}{P(B)} = \frac{\frac{1}{2} - \frac{1}{4}}{\frac{1}{2}} = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2} \]

#### Find \( P(A | B') \):
\[ P(A | B') = \frac{P(A \cap B')}{P(B')} = \frac{P(A) - P(A \cap B)}{P(B')} \]
Calculate \( P(B') \):
\[ P(B') = 1 - P(B) = 1 - \frac{1}{2} = \frac{1}{2} \]
Now substitute:
\[ P(A | B') = \frac{\frac{3}{8} - \frac{1}{4}}{\frac{1}{2}} = \frac{\frac{3}{8} - \frac{2}{8}}{\frac{1}{2}} = \frac{\frac{1}{8}}{\frac{1}{2}} = \frac{1}{4} \]
Now check:
\[ P(A' | B) = 2 \cdot P(A | B') \Rightarrow \frac{1}{2} = 2 \cdot \frac{1}{4} \Rightarrow \frac{1}{2} = \frac{1}{2} \]
So, Option 1 is **correct**.

### Step 3: Check Option 2
We need to check if:
\[ P(A | B) = P(B) \]
Calculate \( P(A | B) \):
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2} \]
Since \( P(B) = \frac{1}{2} \), Option 2 is **correct**.

### Step 4: Check Option 3
We need to check if:
\[ 15 \cdot P(A' | B') = 8 \cdot P(B | A') \]
#### Find \( P(A' | B') \):
\[ P(A' | B') = \frac{P(A' \cap B')}{P(B')} = \frac{1 - P(A \cup B)}{P(B')} = \frac{1 - \frac{5}{8}}{\frac{1}{2}} = \frac{\frac{3}{8}}{\frac{1}{2}} = \frac{3}{4} \]
Now calculate LHS:
\[ LHS = 15 \cdot \frac{3}{4} = \frac{45}{4} \]

#### Find \( P(B | A') \):
\[ P(B | A') = \frac{P(B \cap A')}{P(A')} = \frac{P(B) - P(A \cap B)}{P(A')} \]
Calculate \( P(A') \):
\[ P(A') = 1 - P(A) = 1 - \frac{3}{8} = \frac{5}{8} \]
Now substitute:
\[ P(B | A') = \frac{\frac{1}{2} - \frac{1}{4}}{\frac{5}{8}} = \frac{\frac{1}{4}}{\frac{5}{8}} = \frac{2}{5} \]
Now calculate RHS:
\[ RHS = 8 \cdot \frac{2}{5} = \frac{16}{5} \]
Since \( \frac{45}{4} \neq \frac{16}{5} \), Option 3 is **incorrect**.

### Step 5: Check Option 4
We need to check if:
\[ P(A | B') = P(A \cap B) \]
We already found:
\[ P(A | B') = \frac{1}{4} \quad \text{and} \quad P(A \cap B) = \frac{1}{4} \]
Thus, Option 4 is **correct**.

### Summary of Results
- Option 1: Correct
- Option 2: Correct
- Option 3: Incorrect
- Option 4: Correct