An iterated sequence of Bernoulli trials is carried out and the success probability estimated at each point on the sequence by the current success ratio. We find the probability P 1 that this estimate always lies above some pre-selected rational fraction p′, and its complement P 2, the probability that it will reach p′ or below at least once. In the region p′ ≧ p, P 1 = 0. In the region p′ < p, P 1 ≠ 0 and is furthermore a discontinuous function of p′ at every rational p′.