# Maximizing a Probability Function

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

#### The Case 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

#### The Case 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

#### The Case 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

#### The Case 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

#### Conclusion

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.

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