I'm assuming you meant to say
P(A) = 2/3
P(A and B) = 1/3
If that is the case, then A and B are independent if and only if the following equation is true
P(A and B) = P(A)*P(B)
So we multiply P(A) and P(B) to get the value of P(A and B). We don't know what P(B) is, but we can use algebra to find it
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P(A and B) = P(A)*P(B)
P(A)*P(B) = P(A and B)
(2/3)*P(B) = 1/3
P(B) = (1/3)*(3/2) .... multiply both sides by the reciprocal of 2/3
P(B) = (1*3)/(3*2)
P(B) = 1/2 is the answer
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If P(B) = 1/2, then
P(A and B) = P(A)*P(B)
P(A and B) = (2/3)*(1/2)
P(A and B) = (2*1)/(3*2)
P(A and B) = 1/3
Which is the given probability for both events happening. This confirms we have the correct P(B) value.