A long-standing puzzle in the study of black hole accretion concerns the presence or not of thermal instability. Classical theory predicts that the encircling accretion disc is unstable, as do some self-consistent magnetohydrodynamic simulations of the flow. Yet observations of strongly accreting sources generally fail to exhibit cyclic or unstable dynamics on the expected time-scales. This paper checks whether turbulent fluctuations impede thermal instability. It also asks if it makes sense to conduct linear stability analyses on a turbulent background. These issues are explored with a set of MRI simulations in thermally unstable local boxes in combination with stochastic equations that approximate the disc energetics. These models show that the disc’s thermal behaviour deviates significantly from laminar theory, though ultimately a thermal runaway does occur. We find that the disc temperature evolves as a biased random walk, rather than increasing exponentially, and thus generates a broad spread of outcomes, with instability often delayed for several thermal times. We construct a statistical theory that describes some of this behaviour, emphasizing the importance of the ‘escape time’ and its associated probability distribution. In conclusion, turbulent fluctuations on their own cannot stabilize a disc, but they can weaken and delay thermal instability.