 Continuous Bernoulli priors and certainty

With a uniform prior distribution on a Bernoulli chance $\theta$, $n$ successes yield posterior density $(n+1){\theta }^n$. Integration gives next-trial success probability $(n+1)/(n+2)$, but the singleton $\theta =1$ retains probability zero. More generally, ordinary conditioning with a finite integrable likelihood and positive finite normalization constant preserves absolute continuity, so a continuous prior does not acquire a new atom at certainty from a finite sample.