Answer
$\dfrac{1}{16}$
Work Step by Step
With the given spinner divided equally into $4$ colors, then each color has probability of $\dfrac{1}{4}$ from being chosen. In $2$ spins, the probability of getting a blue and then another blue is
\begin{align*}\require{cancel}
P(\text{blue, then blue})&=
\dfrac{1}{4}\cdot\dfrac{1}{4}
\\\\&=
\dfrac{1}{16}
.\end{align*}
Hence, the probability of getting a blue on the first spin and another blue in the second spin is $
\dfrac{1}{16}
.$