Lewis Carroll’s Pillow Problem

Here is one of the Lewis Carroll’s Pillow Problems (problem # 5):

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?

We will use Bayes’ theory to get the required probability.

$\\ P(W other|W taken) = \frac{P(W taken|W other) * P(W other)}{P(W taken|W other) * P(W other) + P(W taken|B other) * P(B other)} \\ = \frac{1*1/2}{1*1/2 + 1/2*1/2}$

= 1/2/[1/2+1/4] = 2/3 = 0.66

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