A and B are two independent events. The probability that both A and B occur is $\frac{1}{6}$ and the probability that neither occurs is $\frac{1}{3}$. Then probability of the two events respectively occurring is

$\frac{1}{2}$ and $\frac{1}{3}$

Here $P(A∩B) =\frac{1}{6} ⇒P(A). P(B) = \frac{1}{6}$, since the events are independent.

Also $P(\bar{A}∩\bar{B}) =\frac{1}{3} ⇒ P(\overline{A∪B})=\frac{1}{3}$

$⇒ P(A ∪ B) =1-\frac{1}{3}=\frac{2}{3}$

$⇒\frac{2}{3}= P(A) + P(B) - P(A∩B) = P(A) + P(B) -\frac{1}{6}$

$⇒ P(A) + P(B) = \frac{5}{6}$

We find that P(A) and P(B) are the roots of the equation

$6x^2 - 5x + 1 = 0 ⇒ x = \frac{1}{2},\frac{1}{3}$

Hence (A) is the correct answer.