POTW15 Solution

POTW #14-15 Boxes Of Marbles
The Minimum Sample Size To Determine The Contents Of Each Box
John Snyder, FSA
November 26, 2013
17:00 EST
Problem
Imagine that you have three boxes, one containing two black marbles, one containing two white marbles, and the third, one black marble and one white marble. The boxes were labeled for their contents –
BB, WW, and BW – but someone has switched the labels so that every box is now incorrectly labeled.
You are allowed to take one marble at a time out of any box, without looking inside, and by this process
of sampling without replacement you are to determine the contents of all three boxes. What is the
smallest number of drawings needed to do this?
Solution
We only need to draw 1 marble to solve the puzzle.
Assume we draw a marble from the box labeled BW. Since we know that every box is mislabeled, the
box labeled BW must actually be either the BB or the WW box. If the marble we drew was black, then
we know that we drew it from the real BB box. Furthermore, the box labeled WW must actually contain
one white and one black marble since every box is labeled incorrectly and we have already identified
the real BB box. Since we now know which box is actually BB and which is actually BW, the remaining
box must be the real WW one.
By similar reasoning we can identify the contents of each box if the marble we draw is white instead of
black.