  We provide a framework to bound the probability that accumulated errors were
never above a given threshold on hybrid systems. Such systems are used for
example to model an aircraft or a nuclear power plant on one side and its
software on the other side. This report contains simple formulas based on
L\'evy's and Markov's inequalities and it presents a formal theory of random
variables with a special focus on producing concrete results. We selected four
very common applications that fit in our framework and cover the common
practices of hybrid systems that evolve for a long time. We compute the number
of bits that remain continuously significant in the first two applications with
a probability of failure around one against a billion, where worst case
analysis considers that no significant bit remains. We are using PVS as such
formal tools force explicit statement of all hypotheses and prevent incorrect
uses of theorems.
