You are calculating the probability of the bag as though it is BBB, BBW and BWW. However, we know two of those are cases are false since the bag can not be all three.
Someone knowing the state of the marbles would calculate the probabilities differently. It would either be 1, 2/3 or 1/3.
Since it is possible to have BB or WW in the bag at the start it should show that a probability of 2/3 with unknown states and a probability of 2/3 with known states are not equal.
The puzzle solution is based upon a trick getting the reader to believe that a 2/3 with three possible combinations BBB, BBW or BWW is equal to the 2/3 from a known BBW.
Adding a black marble is just a distraction. It is the same as saying that since the probability of drawing a black marble from BB, BW and WW is 1/2 then it must be BW since that is also 1/2.
Not sure if it can be explained better... I'm not the best at probability.
