If A and B are two events such that `P(A)=0.6, P(B) =0.2 and P((A)/(B)) =0.5,` then `P((A')/(B'))` equal to

Options

A. `(1)/(10)`
B. `(3)/(10)`
C. `(3)/(8)`
D. `(6)/(7)`

Answer

To solve the problem, we need to find \( P(A'|B') \), where \( A' \) is the complement of event \( A \) and \( B' \) is the complement of event \( B \).

Given:
- \( P(A) = 0.6 \)
- \( P(B) = 0.2 \)
- \( P(A|B) = 0.5 \)

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

Using the formula for conditional probability:

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

We can rearrange this to find \( P(A \cap B) \):

\[ P(A \cap B) = P(A|B) \cdot P(B) \]

Substituting the known values:

\[ P(A \cap B) = 0.5 \cdot 0.2 = 0.1 \]

### Step 2: Calculate \( P(A \cup B) \)

Using the formula for the union of two events:

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

Substituting the known values:

\[ P(A \cup B) = 0.6 + 0.2 - 0.1 = 0.7 \]

### Step 3: Calculate \( P(B') \)

The probability of the complement of \( B \) is:

\[ P(B') = 1 - P(B) = 1 - 0.2 = 0.8 \]

### Step 4: Calculate \( P(A' \cap B') \)

Using the relationship:

\[ P(A' \cap B') = 1 - P(A \cup B) \]

Substituting the value we found for \( P(A \cup B) \):

\[ P(A' \cap B') = 1 - 0.7 = 0.3 \]

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

Using the definition of conditional probability:

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

Substituting the values we found:

\[ P(A'|B') = \frac{0.3}{0.8} = 0.375 \]

### Final Answer

Thus, \( P(A'|B') = 0.375 \).

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