The book Pillow-Problems: Thought Out During Wakeful Hours, by Charles Dodgson, better known by the pseudonym Lewis Carroll, was first published in 1893. It contains 72 problems that Carroll thought of while lying awake at night over the course of a few decades. Carroll's intent was that the puzzler would, like he did, work out the answers to the questions mentally.
This is problem #23 in the book.
Previous problem: Problem #19.
Next problem: Problem #27.

23.

A bag contains 2 counters, each of which is known to be black or white. 2 white and a black are put in, and 2 white and a black drawn out. Then a white is put in, and a white drawn out. What is the chance that it now contains 2 white?

[25/9/87

Answer:

The original chances, as to States of bag, are

for

2W . . . . .

¼;

1W, 1B . . .

½;

2B . . . . .

¼.

∴ the chances, after adding 2W and 1B, are

for

4W, 1B . . . .

¼;

3W, 2B . . . .

½;

2W, 3B . . . .

¼.

Now the chances, which these give to the observed event, drawing 2W and 1B, are

(3)/

()

, [sic; should be

(3)/

(5)

]

(3)/

(5)

,

(3)/

(10)

.

∴ the chances, after this event, are proportional to

(3)/

(20)

,

(3)/

(10)

,

(3)/

(40)

i.e.to 2, 4, 1. Hence they are

(2)/

(7)

,

(4)/

(7)

,

(1)/

(7)

.

Hence the chances, as to States, now are

for

2W . . . . .

(2)/

(7)

;

1W, 1B. . . .

(4)/

(7)

;

2B . . . . .

(1)/

(7)

.

∴ the chances, after adding 1W, are

for

3W . . . . .

(2)/

(7)

;

2W, 1B. . . .

(4)/

(7)

;

1W, 2B. . . .

(1)/

(7)

.

Now the chances, which these give to the observed event, of drawing 1W, are 1,

(2)/

(3)

,

(1)/

(3)

.

∴ the chances, after this event, are proportional to

(2)/

(7)

,

(8)/

(21)

,

(1)/

(21)

; i.e. to 6, 8, 1. Hence they are

(6)/

(15)

,

(8)/

(15)

,

(1)/

(15)

.

Hence the chance, that the bag now contains 2 white, is