Pillow-Problems: Problem #16

Problem #15 | Problem #19
For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll).

Problem:

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?

Answer:

The first course gives chance = ½; the second,

(5)/

(12)

. Hence the first is best.

Solution:

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.