A probabilistic variant of input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, is introduced. It is proved that these automata can be determinized: an n-state probabilistic IDPDA that accepts each string with probability at least λ+δ or at most λ−δ, for some isolated cut-point λ∈[0,1] with δ>0, is transformed to a deterministic IDPDA recognizing the same language, with at most (1+[Formula presented])n2−n states and at most |Σ+1|⋅(1+[Formula presented])n2−n stack symbols, where Σ+1 is the set of left bracket symbols in the alphabet. An asymptotically close lower bound is provided: for infinitely many n, there is a probabilistic IDPDA with 2n+3 states and n stack symbols, and with δ=[Formula presented], such that every equivalent deterministic IDPDA needs at least 7n2/14 states and at least n stack symbols. A few special cases of automata with reduced determinization complexity are identified.