Here is Lewis Carroll’s last pillow problem (1958). A bag contains two counters, as to which nothing is known except that each is either black or white. Ascertain their colors without taking them out of the bag. Carroll’s answer is One is black, and the other is white. What do you think of his “solution”? We know that, if a bag contained three counters, two being black and one white, the chance of drawing a black one would be 2/3; and that any other state of things would not give this chance. Now the chances, that the given bag contains
(a) BB,
(b) BW,
(c) WW, are respectively,1/4, 1/2, 1/4.
Add a black counter. Then the chances that it contains
(a) BBB,
(b) BWB,
(c) WWB are, as before, 1/4, 1/2, 1/4.
Hence the chance of now drawing the black one is 1/4 (1) + 1/2 (2/3) + 1/4 (1/3) = 2/3.
Hence the bag now contains BBW (since any other state of things would not give this chance). Hence, before the black counter was added, it contained BW, i.e., one black counter and one white.
