We investigate the Hölder regularity of the function $T$ of the probability of tending to one minimal set, the partial derivatives of $T$ with respect to the probability parameters, which can be regarded as complex analogues of the Takagi function, and the higher partial derivatives $C$ of $T.$ Our main result gives a dynamical description of the pointwise Hölder exponents of $T$ and $C$, which allows us to determine the spectrum of pointwise Hölder exponents by employing the multifractal formalism in ergodic theory. Also, we prove that the bottom of the spectrum $α_{-}$ is strictly less than $1$, which allows us to show that the averaged system acts chaotically on the Banach space $C^{α}$ of $α$- Hölder continuous functions for every $α\in (α_{-},1)$, though the averaged system behaves very mildly (e.g. we have spectral gaps) on $C^{β}$ for small $β>0.$