## Pillow-Problems: #16

Difficulty: (explanation of difficulty)

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 #16 in the book.
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### 16.

There are two bags, one containing a counter, known to be either white or black; the other containing 1 white and 2 black. A white is put into the first, the bag shaken, and a counter drawn out, which proves to be white. Which course will now give the best chance of drawing a white—to draw from one of the two bags without knowing which it is, or to empty one bag into the other and then draw?

[10/87

Draw from one of the two bags without knowing which.
Empty one bag into the other and then draw.

The ‘a priori’ chances of possible states of first bag are ‘W, ½ B, ½’. Hence chances, after putting W in, are ‘WW, ½ WB, ½’. The3 chances, which these give to the ‘observed event’, are 1, ½. Hence chances of possible states ‘W, B’, after the event, are proportional to 1, ½ i.e. to 2, 1; i.e. their actual values are

 (2)/ (3)
,
 (1)/ (3)
.

Now, in first course, chance of drawing W is ½ ·

 (2)/ (3)
+ ½ & middot;
 (1)/ (3)
; i.e. ½.

And, in second course, chances of possible states ‘WWBB, WBBB’ are

 (2)/ (3)
,
 (1)/ (3)
: hence chance of drawing W is
 (2)/ (3)
· ½ +
 (1)/ (3)
· ¼; i.e.
 (5)/ (12)
.

Hence first course gives best chance.

Q.E.F.