If A and B are two independent events with `P(A) = (3)/(5)` and `P(B) = (4)/(9)` , then `P(A' cap B')` equal to :

Options

A. `4//15`
B. `8//45`
C. `1//3`
D. `2//9`

Answer

To find \( P(A' \cap B') \) where \( P(A) = \frac{3}{5} \) and \( P(B) = \frac{4}{9} \), we can follow these steps:

### Step 1: Understand the relationship between events
Since \( A \) and \( B \) are independent events, we know that:
\[ P(A \cap B) = P(A) \times P(B) \]

### Step 2: Calculate \( P(A \cap B) \)
Substituting the values:
\[ P(A \cap B) = P(A) \times P(B) = \frac{3}{5} \times \frac{4}{9} \]
Calculating this gives:
\[ P(A \cap B) = \frac{3 \times 4}{5 \times 9} = \frac{12}{45} \]

### Step 3: Use the formula for the union of events
To find \( P(A' \cap B') \), we can use the formula:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
Where:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 4: Calculate \( P(A \cup B) \)
Substituting the known values:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{3}{5} + \frac{4}{9} - \frac{12}{45} \]

### Step 5: Find a common denominator
The least common multiple of 5, 9, and 45 is 45. We convert each term:
\[ P(A) = \frac{3}{5} = \frac{27}{45} \]
\[ P(B) = \frac{4}{9} = \frac{20}{45} \]
Now substituting these values:
\[ P(A \cup B) = \frac{27}{45} + \frac{20}{45} - \frac{12}{45} = \frac{27 + 20 - 12}{45} = \frac{35}{45} \]

### Step 6: Calculate \( P(A' \cap B') \)
Now we can find:
\[ P(A' \cap B') = 1 - P(A \cup B) = 1 - \frac{35}{45} = \frac{10}{45} \]
Reducing this fraction gives:
\[ P(A' \cap B') = \frac{2}{9} \]

### Final Answer
Thus, the probability \( P(A' \cap B') \) is:
\[ \boxed{\frac{2}{9}} \]