Lewis Carroll's Pillow Problem

This problem is cited by M. Gardner in his Mathematical Circus and also Gardner's Workout.

A bag contains a counter, known to be either white or black. A white counter is put in, the bag is shaken, and a counter is drawn out, which proves to be white. What is now the chance of drawing a white counter?

Lewis Carroll offers two solutions - one short and wrong (Proof 1 below), another right but long. (He of course new which is which.) Martin Gardner [Circus, p. 189] calls the second of Carroll's proofs "long-winged" and affers instead a shorter one (Proof 2 below) by one of his readers, Howard Ellis from Chicago.

Solution #1

As the state of the bag, after the operation, is necessarily identical with its state before it, the chance is just what it was, viz. 1/2.

Solution #2

Let B and W1 stand for the black or white counter that may be in the bag at the start and W2 for the added white counter. After removing white counter there are three equally likely states:

Inside bag	Outside bag
W1	W2
W2	W1
B	W2

In two of these states a white counter remains in the bag, and so the chance of drawing a white counter the second time is 2/3.

References

1. M. Gardner, Gardner's Workout, A K Peters, 2001, pp. 129-132
2. M. Gardner, Mathematical Circus, Vintage, 1981