A "Pillow Problem" by Lewis Carroll (1895)


 

 

Charles Dodgson (1832-1898), whose pen name was Lewis Carroll, was an English author, mathematician, logician, photographer, and church deacon best known for writing the books Alice's Adventures in Wonderland and Through the Looking-Glass. He wrote a book of 72 mathematics problems, titled Pillow Problems thought out during Wakeful Hours, which he said he had personally worked out without putting pen to paper, and here is problem 5:

 

A bag contains one counter, known to be either white or black. A white counter is put in, the bag shaken, and a counter drawn out, which proves to be white. What is now the chance of drawing a white counter?

Additional information

The person who put the first token into the bag had only two tokens, one white and one black. He flipped a coin to decide which one to keep and which one to put in the bag.

 

Mr. Dodgson suggested that you should try to solve this riddle in your head before you reach for paper and pencil.

 

 

 

 

Here is a detailed solution that is obtained by listing all possible cases

 

Before a white token was drawn from the bag, the bag could have contained either two different white tokens, W, W, or one black and one white token, B, W. Both cases were equally probable, (probability 1/2), because the color of the first token was decided by the flip of a coin.

Therefore, before the drawing, there were four equally probable outcomes of drawing one token from the bag:

 

Case

Probability

Outcome

Color

Probability

Token left

W W

1/2

first token

W

1/2 * 1/2 = 1/4

W

 

 

second token

W

1/2 * 1/2 = 1/4

W

B W

1/2

first token*

B

1/2 * 1/2 = 1/4

W

 

 

second token

W

1/2 * 1/2 = 1/4

B

 

But we know that it was a white token that was drawn from the bag. Therefore, the third case above, marked *, has been excluded by the drawing. And we are left with only three equally probable cases of the contents of the bag:

 

Token left

W

W

(impossible)

B

 

So the chance that the remaining token is white is two thirds.


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