# Pillow-Problems: Problem #16

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## Problem:

16. (20, 40)

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

16. (4, 40)

The first course gives chance = ½; the second,
 (5)/ (12)
. Hence the first is best.

## Solution:

16. (4, 20)

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 ·
 (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.