conditional probability:

if you think this in conditional probebility way:

P(A|B) = P(A&B)/P(B)

event A: the remaining marble from the same bag is also white

event B: the first picking marble is white

P(B) = 1/3*1 + 1/3*1/2 = 1/2 

P(A&B) = 1/3 (the event A&B means that you must pick the "two white marble" bag, this happens with probability 1/3)

so P(A|B) = 2/3