We consider transitions to chaos in random dynamical systems induced by an increase in noise amplitude. We show how the emergence of chaos (indicated by a positive Lyapunov exponent) in a logistic map with bounded additive noise can be analyzed in the framework of conditioned random dynamics through expected escape times and conditioned Lyapunov exponents for a compartmental model representing the competition between contracting and expanding behavior. In contrast to the existing literature, our approach does not rely on small noise assumptions, nor does it refer to deterministic paradigms. We find that the noise-induced transition to chaos is caused by a rapid decay of the expected escape time from the contracting compartment, while all other order parameters remain approximately constant.