Download Solution to Problem of the Week Five

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The problems of the week are available online at
http://www.math.jmu.edu/∼rosenhjd/POTW.html.
Solution to Problem of the Week Five
A box contains 999, 999 white marbles and 1, 000, 001 black marbles.
You also have an essentially endless pile of black marbles outside
of the box. You reach into the box and remove two marbles at
random. If they are of different colors, you place the white marble
back in the box and discard the black one. If they are of the same
color, you remove both of them and place one black marble from
your pile into the box. Notice that each time you repeat this
process, the number of marbles in the box goes down by one.
Continue this process until only one marble remains in the box.
What is the probability that the one remaining marble is black?
The probability is zero that the one remaining marble is black. Notice
that white marbles only leave the box in pairs. That means that each time
you repeat the process the number of white marbles in the box either remains
the same or decreases by two. Since there are an odd number of white marbles
in the box to start with, it follows that the number of white marbles in the
box will always be odd. Consequently, you will never be able to remove the
last white marble from the box. When only one marble remains, it has to be
white.