In this paper a fixed point theorem for a probabilistic $q$-contraction $f:S\to S,$ where $(S, {\mathcal{F}},T)$ is a complete Menger space, ${\mathcal{F}}$ satisfies a grow condition, and $T$ is a $g$-convergent t-norm (not necessarily $T \geq T_{\mathbf{L}}$) is proved. There is proved also a second fixed point theorem for mappings $f:S \rightarrow S$, where $(S, {\mathcal{F}},T)$ is a complete Menger space, ${\mathcal{F}}$ satisfy a weaker condition than in [V. Radu: Lectures on probabilistic analysis. Surveys. (Lectures Notes and Monographs Series on Probability, Statistics \& Applied Mathematics 2), Universitatea de Vest din Timişoara 1994.], and $T$ belongs to some subclasses of Dombi, Aczél-Alsina, and Sugeno-Weber families of t-norms. An application to random operator equations is obtained.