Answer
$\color{blue}{\dfrac{13}{64}}$
Work Step by Step
$\begin{align*}
P\left(\left\vert Y - \frac{1}{2}\right\vert\lt \frac{1}{4}\right) &= P\left(-\frac{1}{4}\lt Y - \frac{1}{2}\lt \frac{1}{4}\right) \\
&= P\left(\frac{1}{2}-\frac{1}{4}\lt Y \lt \frac{1}{2} + \frac{1}{4}\right) \\
&= P\left( \frac{1}{4} \lt Y \lt \frac{3}{4} \right) \\
&= \int_{1/4}^{3/4} \frac{3}{2}y^2\ dy \\
&= \left[ \frac{3}{2}\frac{y^3}{3} \right]_{1/4}^{3/4} \\
&= \left[ \frac{1}{2}y^3 \right]_{1/4}^{3/4} \\
&= \frac{1}{2}((3/4)^3 - (1/4)^3) \\
&= \frac{1}{2}(27/64 - 1/64) \\
&= \frac{1}{2}\cdot \frac{26}{64} \\
\color{blue}{P\left(\left\vert Y - \frac{1}{2}\right\vert\lt \frac{1}{4}\right)} &\color{blue}{= \frac{13}{64}}
\end{align*}$
