Half of 2 of anything is 1 of that thing. If the thing happens to be a "third", then half of 2 thirds is 1 third.
(1/2)*(2/3) = (1*2)/(2*3) = (1*2)/(3*2) = (1/3)*(2/2) = (1/3)*1 = 1/3
(1/2)*(2/3) = (1*2)/(2*3) = (1*2)/(3*2) = (1/3)*(2/2) = (1/3)*1 = 1/3