Gibbs-type priors are arguably the most 'natural' generalization of the Dirichlet prior. Among them the two parameter Poisson-Dirichlet prior certainly stands out for the simplicity and intuitiveness of its predictive probabilities. Given an observable sample of size $n$, in this paper we show that the predictive probabilities of any Gibbs-type prior admit a large $n$ approximation, with an error term vanishing as $o(1/n)$, which maintains the same mathematical tractability and interpretability as the predictive probabilities of the two parameter Poisson-Dirichlet prior. We discuss the use of our approximate predictive probabilities in connection with some recent work on Bayesian nonparametric inference for discovery probabilities.