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 first excited state, n=2, the normalized wave function $\psi=\sqrt{\frac{2}{L}}sin(2 \pi x/L)$.
a. $\frac{2}{L}sin^2(\pi/2)dx=2 dx/L$
b. $\frac{2}{L}sin^2(\pi)dx=0$
c. $\frac{2}{L}sin^2(3\pi/2)dx=2 dx/L$
Our results agree with Figure 40.12b in the textbook. The probability density is zero at the center of the box and is symmetric about the center of the box.