The stochastic Hindmarsh-Rose model is studied in the parameter region where two bursting limit cycles of different types coexist. We show that under the influence of noise, transitions between basins of attractions appear, which generates stochastic bursting oscillations of mixed modes. The formation of this new regime is accompanied by anti-coherence and coherence resonances as well as by the transition to chaos. We investigate the probabilistic mechanism of the noise-driven bursting birhythmicity using the stochastic sensitivity functions and confidence domains method. © 2023 Author(s).