For two events. A and B. it is given that `P(A)= (3)/(5), P(B)= (3)/(10)` and `P(A|B)= (2)/(3)`. IF `barA` and `barB` are the complementary events of A and B. then what is `P(barA|barB)` equal to ?

Options

A. `(3)/(7)`
B. `(3)/(4)`
C. `(1)/(3)`
D. `(4)/(7)`

Answer

To solve the problem, we need to find \( P(\bar{A} | \bar{B}) \), where \( \bar{A} \) and \( \bar{B} \) are the complements of events \( A \) and \( B \) respectively. We are given:

- \( P(A) = \frac{3}{5} \)
- \( P(B) = \frac{3}{10} \)
- \( P(A|B) = \frac{2}{3} \)

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

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:

\[ P(A \cap B) = \frac{2}{3} \cdot \frac{3}{10} = \frac{2 \cdot 3}{3 \cdot 10} = \frac{2}{10} = \frac{1}{5} \]

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

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:

\[ P(A \cup B) = \frac{3}{5} + \frac{3}{10} - \frac{1}{5} \]

To perform this calculation, we need a common denominator. The least common multiple of 5 and 10 is 10:

\[ P(A \cup B) = \frac{6}{10} + \frac{3}{10} - \frac{2}{10} = \frac{6 + 3 - 2}{10} = \frac{7}{10} \]

### Step 3: Find \( P(\bar{A}) \) and \( P(\bar{B}) \)

Now we can find the probabilities of the complements:

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

\[ P(\bar{B}) = 1 - P(B) = 1 - \frac{3}{10} = \frac{7}{10} \]

### Step 4: Find \( P(\bar{A} \cap \bar{B}) \)

Using the relationship:

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

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

\[ P(\bar{A} \cap \bar{B}) = 1 - \frac{7}{10} = \frac{3}{10} \]

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

Using the conditional probability formula:

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

Substituting the values we have:

\[ P(\bar{A} | \bar{B}) = \frac{\frac{3}{10}}{\frac{7}{10}} = \frac{3}{7} \]

### Final Answer

Thus, the probability \( P(\bar{A} | \bar{B}) \) is:

\[ \boxed{\frac{3}{7}} \]