Nothing is wrong with the calculation, but its interpretation is nonsense. When the various probabilities are combined, we are working out the probability of extracting a black rabbit, over *all possible combinations of rabbits*. It is fallacious to imagine that this probability is valid for any specific combination. The fallacy is glaring if there is only one rabbit in the hat. With one rabbit, a similar argument (ignoring adding and removing a black rabbit, which changes nothing essential) goes like this: the hat contains either B or W, each with probability .

The probability of extracting a black rabbit is therefore

which is . Therefore (*really?*) half the rabbits in the hat are black, and half are white.

But there's only one rabbit in the hat...

*As seen on page 19, Professor Stewart's Cabinet of Mathematical Curiosities*