Suppose A and B are two events such that `P(A)=1/4, P(B)=1/3` and `P(AnnB)=1/5`, then 4P(A'|B') is equal to ____

Answer

To solve the problem, we need to find \( 4P(A'|B') \) given the probabilities \( P(A) = \frac{1}{4} \), \( P(B) = \frac{1}{3} \), and \( P(A \cap B) = \frac{1}{5} \).

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

Using the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the given values:
\[ P(A \cup B) = \frac{1}{4} + \frac{1}{3} - \frac{1}{5} \]
To perform this calculation, we need a common denominator. The least common multiple of 4, 3, and 5 is 60.

Converting each fraction:
\[ P(A) = \frac{1}{4} = \frac{15}{60}, \quad P(B) = \frac{1}{3} = \frac{20}{60}, \quad P(A \cap B) = \frac{1}{5} = \frac{12}{60} \]

Now substituting these values:
\[ P(A \cup B) = \frac{15}{60} + \frac{20}{60} - \frac{12}{60} = \frac{15 + 20 - 12}{60} = \frac{23}{60} \]

### Step 2: Calculate \( P(A' \cap B') \)

Using De Morgan's law, we know:
\[ P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B) \]
Substituting the value we found:
\[ P(A' \cap B') = 1 - \frac{23}{60} = \frac{60 - 23}{60} = \frac{37}{60} \]

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

Using the complement rule:
\[ P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \]

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

Using the definition of conditional probability:
\[ P(A' | B') = \frac{P(A' \cap B')}{P(B')} \]
Substituting the values we have:
\[ P(A' | B') = \frac{\frac{37}{60}}{\frac{2}{3}} = \frac{37}{60} \times \frac{3}{2} = \frac{37 \times 3}{60 \times 2} = \frac{111}{120} = \frac{37}{40} \]

### Step 5: Calculate \( 4P(A' | B') \)

Finally, we need to find:
\[ 4P(A' | B') = 4 \times \frac{37}{40} = \frac{148}{40} = \frac{37}{10} = 3.7 \]

Thus, the final answer is:
\[ \boxed{3.7} \]