Quantum Chemistry/Example 2

Question

Starting with the wave equation of a 1D box:

$\mathrm{\Psi }\left(x\right)=\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi }{L}x\right)$

The classical probability of a particle existing in the region of the box [0, 1/L] is,

$P{(1/L)}_{classical}=\frac{1}{L}$

Where $L$ is the length of the 1D box, $n$ is the principle quantum number, and $x$ is the position of the particle in a 1D box.

Derive an equation that calculates the probability of the particle being in the [0, 1/L] for the quantum mechanical particle in a 1D box. Is this result compatible with the correspondence principle?