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We analyse the asymptotic behaviour of the probability of observing theexpected number of successes at each stage of a sequence of nested Bernoullitrials. Our motivation is the attempt to give a genuinely frequentistinterpretation to the notion of probability based on finite sample sizes. Themain result is that the probabilities under consideration decay asymptoticallyas $n^{-1/3}$, where $n$ is the common length of the Bernoulli trials. The mainingredient in the proof is a new fixed-point theorem for non-contractivesymmetric functions of the unit interval.

On the expected number of successes in a sequence of nested Bernoulli trials