  Many phenomena in the climate system lie in the gray zone between weather and
climate: they are not amenable to deterministic forecast, but they still depend
on the initial condition. A natural example is medium-range forecasting, which
is inherently probabilistic because it lies beyond the deterministic
predictability time of the atmosphere, but for which statistically significant
prediction can be made which depend on the current state of the system.
Similarly, one may ask the probability of occurrence of an El Ni\~no event
several months ahead of time. In this paper, we introduce a quantity which
corresponds precisely to this type of prediction problem: the committor
function is the probability that an event takes place within a given time
window, as a function of the initial condition. We explain the main
mathematical properties of this probabilistic concept, and compute it in the
case of a low-dimensional stochastic model for El-Ni\~no, the Jin and Timmerman
model. In this context, we show that the ability to predict the probability of
occurrence of the event of interest may differ strongly depending on the
initial state. The main result is the new distinction between intrinsic
probabilistic predictability (when the committor function is smooth and
probability can be computed which does not depend sensitively on the initial
condition) and intrinsic probabilistic unpredictability (when the committor
function depends sensitively on the initial condition). We also demonstrate
that the Jin and Timmerman model might be the first example of a stochastic
differential equation with weak noise for which transition between attractors
do not follow the Arrhenius law, which is expected based on large deviation
theory and generic hypothesis.
