Let A and B be two events such that P(A) = 1/3, P(B) = 1/4 and P( A `nn`B ) = 1/5, then 45P(A' |B') - 35.32 is equal to _____

Answer

To solve the problem, we need to find the value of \( 45P(A' | B') - 35.32 \) given the probabilities of events \( A \) and \( B \) and their intersection.

### Step-by-Step Solution:

1. **Identify Given Probabilities:**
- \( P(A) = \frac{1}{3} \)
- \( P(B) = \frac{1}{4} \)
- \( P(A \cap B) = \frac{1}{5} \)

2. **Calculate \( P(A \cup B) \):**
Using the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substitute the known values:
\[ P(A \cup B) = \frac{1}{3} + \frac{1}{4} - \frac{1}{5} \]
To perform this calculation, find a common denominator (which is 60):
\[ P(A \cup B) = \frac{20}{60} + \frac{15}{60} - \frac{12}{60} = \frac{20 + 15 - 12}{60} = \frac{23}{60} \]

3. **Calculate \( P(A' \cap B') \):**
Using De Morgan's Law:
\[ P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B) \]
Substitute the value calculated:
\[ P(A' \cap B') = 1 - \frac{23}{60} = \frac{60 - 23}{60} = \frac{37}{60} \]

4. **Calculate \( P(B') \):**
\[ P(B') = 1 - P(B) = 1 - \frac{1}{4} = \frac{3}{4} \]

5. **Calculate \( P(A' | B') \):**
Using the formula for conditional probability:
\[ P(A' | B') = \frac{P(A' \cap B')}{P(B')} \]
Substitute the values:
\[ P(A' | B') = \frac{\frac{37}{60}}{\frac{3}{4}} = \frac{37}{60} \times \frac{4}{3} = \frac{37 \times 4}{60 \times 3} = \frac{148}{180} = \frac{37}{45} \]

6. **Calculate the final expression:**
\[ 45P(A' | B') - 35.32 = 45 \times \frac{37}{45} - 35.32 = 37 - 35.32 = 1.68 \]

### Final Answer:
The value of \( 45P(A' | B') - 35.32 \) is \( \boxed{1.68} \).