  We analyse the asymptotic behaviour of the probability of observing the
expected number of successes at each stage of a sequence of nested Bernoulli
trials. Our motivation is the attempt to give a genuinely frequentist
interpretation to the notion of probability based on finite sample sizes. The
main result is that the probabilities under consideration decay asymptotically
as $n^{-1/3}$, where $n$ is the common length of the Bernoulli trials. The main
ingredient in the proof is a new fixed-point theorem for non-contractive
symmetric functions of the unit interval.
