For a finitely presented group \( \Gamma \) with a finite presentation, let \( X_\Gamma \) be the presentation 2-complex. We introduce \( n \)-fold random branched coverings of \( X_\Gamma \) branched over its 2-cells. With probabilistic notions, we prove that fundamental groups of random branched coverings are asymptotically almost surely Gromov hyperbolic. In other words, for a random branched covering \( \overline{X}_\Gamma \rightarrow X_\Gamma \), the probability that \( \pi_1(X_\Gamma) \) is Gromov hyperbolic goes to 1 in the limit \( n \rightarrow \infty \).