Quantum Chemistry/Example 2

Question
Starting with the wave equation of a 1D box:

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

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

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

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?