If A and B be two events such that `P(A)=1//4,` `P(B)=1//3` and `P(AuuB)=1//2` show that A and B are independent events.

Explanation

For independent events A and B,` P(A cap B)=P(A) P(B)`.


Also, `P(A cup B)=P(A)+P(B)-P(A cap B`)


Put the given values,


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


`frac{1}{2}=frac{7}{12}-P(A cap B)`


`Rightarrow P(A cap B)=frac{7}{12}-frac{1}{2}=frac{1}{12} `


`Rightarrow P(A cap B)=frac{1}{4} \times\ frac{1}{3}=P(A) P(B)`


Thus, A and B are independent events.