  In this paper, we investigate the computational and verification power of
bounded-error postselecting realtime probabilistic finite state automata
(PostPFAs). We show that PostPFAs using rational-valued transitions can do
different variants of equality checks and they can verify some nonregular unary
languages. Then, we allow them to use real-valued transitions (magic-coins) and
show that they can recognize uncountably many binary languages by help of a
counter and verify uncountably many unary languages by help of a prover. We
also present some corollaries on probabilistic counter automata.
