Coin tossing, revisited

An iterated sequence of Bernoulli trials is carried out and the success probability estimated at each point on the sequence by the current sucdess ratio. We find the probability P¹ that this estimate always lies above some pre-selected rational fraction p', and its complement P², the probability that it will reach p' or below at least once. In the region p'≥p, P¹=0. In the region p'<p, P¹=ab42/0 and is furthermore a discontinuous function of p' at every rational p'.