What do you think of the following proof by Lewis Carroll that an urn cannot contain two balls

of the same colour? Suppose that the urn contains two balls, each of which is either black or white;

thus, in the obvious notation, P(BB) = P(BW) = P(WB) = P(WW) = 1

4 . We add a black ball, so

that P(BBB) = P(BBW) = P(BWB) = P(BWW) = 1

4 . Next we pick a ball at random; the chance

that the ball is black is (using conditional probabilities) 1 1

4 + 23

14

+ 2

3 1

4 + 1

3 14= 23 . However, ifthere is probability 23that a ball, chosen randomly from three, is black, then there must be two blackand one white, which is to say that originally there was one black and one white ball in the urn