Extracted from the book "Exploring Mathmematics with Scientific Notebook", Published by Springer Verlag, ISBN #981-3083-88-3, by me and Professor Jonathan Lewin of Kennesaw State University, Georgia.
In this example, we suppose that we have
white balls and
black balls which we are going to place in two urns A and B in any way we
please, as long as at least one ball is placed into each urn. After this has
been done, a second person walks into the room and selects one ball at random.
Our problem is to maximize the probability that this person draws a white
ball.
We suppose that the distribution of the balls in the urns A and B is as
described in the following table:
If
is the probability that a single ball drawn at random will be white then
From now on we shall assume that
We begin our study of the function by looking at the following table which
shows the values of
at a few selected points
To solve the problem we need to find the maximum value of the expression
as the point
varies through the rectangle
from which the points
and
have been removed. If we sketch the graph
then we obtain the following surface:
From the looks of this surface it seems unlikely that the maximum value of
will be achieved at a critical point. The maximum appears to be at the left or
right extremities of the figure. As a matter of fact, if we point at the
equations
and click on Solve and Exact
then we obtain
As we have already seen, the maximum value of
does not occur at the point
We now examine the boundary behavior of the function. There are four cases to consider
and 
We define
for
Point at this definition of
and click on Define and New
Definition. Since
for each
we see that the maximum value of
is
and 
We define
for
Point at this definition of
and click on Define and New
Definition. Since
for each
we see that the maximum value of this function is
and 
We define
for
Point at this definition of
and click on Define and New
Definition. Since
for each
we see that the maximum value of this function is
and 
We define
for
Point at this definition of
and click on Define and New
Definition. Since
for each
we see that the maximum value of this function is
We conclude that the expression
takes a maximum value of
at the point
and again at the point
This means that we can maximize the probability that a white ball will be
selected by placing one white ball and no black ball in urn A and all the
other balls in urn B. Alternatively we can place one white ball and just no
black ball in urn B and all the other balls in urn A.