​$$\text{For two events A and B, } {P(A)=\frac{1}{3}}, {P(B)=\frac{1}{4}} \text{ and } {P(A \cup B) = \frac{1}{2}}. \ \text{find} \ P( \frac {A}{B})=?$$

Correct option is D

Given: 

​$$P(A) = \frac{1}{3}, \quad P(B) = \frac{1}{4}, \quad P(A \cup B) = \frac{1}{2}$$​ 

Formula Used:

​$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$​ 

​$$P(A | B) = \frac{P(A \cap B)}{P(B)}$$ 

Solution: 

$$P(A \cap B) = \frac{1}{3} + \frac{1}{4} - \frac{1}{2}$$ 

$$= \frac{4}{12} + \frac{3}{12} - \frac{6}{12}$$  = $$\frac1{12}$$ 

Then; 

$$P(A | B)$$ 

$$= \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{12}}{\frac{1}{4}}$$ 

$$= \frac{1}{12} \times \frac{4}{1} = \frac{4}{12} = \frac{1}{3}$$