There is a Russian book which is supposed to be a translation Lewis Carroll's, collection's of math problems. (The problems are collected from SEVERAL books and maybe from manuscripts of Carroll, and I could never find the English source of the following problem which seems expecially interesting to me).

(This seems to confirm the general rule stated by one Russian poet that there are more English verses and tales in Russian than in English, this general rule seems to apply to problems as well:-)

It is under the heading:

TRANSCENDENTAL PROBABILITY THEORY.

Problem. An urn contains 2 balls which can be black or white. (Someone put them in advance into the urn randomly, with equal probability of black and white). Find the colors of the balls without taking them out of the urn:-)

Answer. One black and one white.

Proof. 1. Let us ADD to the urn one additional black ball. And consider the probability that a ball taken randomly from the urn NOW will be black. This probability is 2/3 IF AND ONLY IF the urn originally contained ONE BLACK AND ONE WHITE ball. (Otherwise it is 1/3 or 1)

2. Now let us compute this probability (that a ball randomly taken from the urn with three balls is black). The probabilities before we added the black ball were: 

	      WW with probability 1/4,
	      WB with probability 1/2, and
	      BB with probability 1/4.

Now, after we put the black ball, the probability that a ball randomly taken from the urn will be black is:

(1/3)(1/4)+(2/3)(1/2)+1(1/4)=2/3,

so, according to step 1, the urn contains 2 black balls and one white, and this proves that the original urn contained one black and one white.

Notice we did not TAKE any balls from the urn, we only ADDED one, and then computed probabilities!

Where is the mistake in Carroll's "solution"?