It is unknown how abundant extraterrestrial life is, or whether such life might be complex or intelligent. On Earth, the emergence of complex intelligent life required a preceding series of evolutionary transitions such as abiogenesis, eukaryogenesis, and the evolution of sexual reproduction, multicellularity, and intelligence itself. Some of these transitions could have been extraordinarily improbable, even in conducive environments. The emergence of intelligent life late in Earth's lifetime is thought to be evidence for a handful of rare evolutionary transitions, but the timing of other evolutionary transitions in the fossil record is yet to be analyzed in a similar framework. Using a simplified Bayesian model that combines uninformative priors and the timing of evolutionary transitions, we demonstrate that expected evolutionary transition times likely exceed the lifetime of Earth, perhaps by many orders of magnitude. Our results corroborate the original argument suggested by Brandon Carter that intelligent life in the Universe is exceptionally rare, assuming that intelligent life elsewhere requires analogous evolutionary transitions. Arriving at the opposite conclusion would require exceptionally conservative priors, evidence for much earlier transitions, multiple instances of transitions, or an alternative model that can explain why evolutionary transitions took hundreds of millions of years without appealing to rare chance events. Although the model is simple, it provides an initial basis for evaluating how varying biological assumptions and fossil record data impact the probability of evolving intelligent life, and also provides a number of testable predictions, such as that some biological paradoxes will remain unresolved and that planets orbiting M dwarf stars are uninhabitable.