Answer
See explanation.
Work Step by Step
The quantity $|\psi|^2$ is a probability distribution function. The desired probability is $|\psi|^2dx$, with $\psi$ evaluated at the point in question.
For the ground state, the normalized wave function $\psi=\sqrt{\frac{2}{L}}sin(\pi x/L)$.
a. $\frac{2}{L}sin^2(\pi/4)dx=dx/L$
b. $\frac{2}{L}sin^2(\pi/2)dx=2dx/L$
c. $\frac{2}{L}sin^2(3\pi/4)dx=dx/L$
Our results agree with Figure 40.12b in the textbook. The probability density is largest at the center of the box and is symmetric about the center of the box.
