There is no paradox. There is only confusion due to a complicated analysis.

Basically, the situation is as follows.
  • 1/4 WWW, P(White) = 1/4 * 1 or 3/12. P(Black) = 1/4 * 0 or 0
  • 1/2 BWW, P(White) = 1/2 * 2/3 or 4 /12 P(Black) = 1/2 * 1/3 or 2/12
  • 1/4 BBW, P(White) = 1/4 * 1/3 or 1/12 P(Black) = 1/4 * 2/3 or 2/12


TotalP(White) = 8/12 or 2/3
TotalP(black) = 4/12 or 1/3

The above is a correct analysis, and agrees with your computation of probabilitites.

The paradox is resolved by noting that the following statment is not valid.
Again, on the other direction: if the probability to select a white ball is 2/3, the situation MUST be WWB.