Answer
The probability distribution function would be uniform. Yes.
Work Step by Step
A classical particle would have an equal probability of being found anywhere within the box, because it would move at constant speed and bounce elastically off the 2 walls. The probability distribution function would be constant inside the box.
As the particle energy rises, and n increases, the wavefunctions have more sine waves, and more and more peaks. See figure 40.12a.
The probability distribution function will indeed approach a uniform value as n approaches infinity. See figure 40.12b. The integral between any two points within the box will only depend upon the distance between them, not on their location.
