Among the host of answers supplied, I would also call attention to Bayesian statistics. Some problems cannot be answered by likelihoods alone. A Frequentist uses counterfactual reasoning where the "probability" refers to alternate universes and an alternate universe framework makes no sense as far as inferring the state of an individual, such as the guilt or innocence of a criminal, or whether bottlenecking of gene frequency in a species exposed to a massive environmental shift led to its extinction. In the Bayesian context, probability is "belief" not frequency, which can be applied to that which has already precipitated.
Now, the majority of Bayesian methods require fully specifying probability models for the prior and the outcome. And, most of these probability models are parametric. Consistent with what others are saying, these need not be exactly correct to produce meaningful summaries of the data. "All models are wrong, some models are useful."
There is, of course, nonparametric Bayesian methods. These have a lot of statistical wrinkles and, generally speaking, require nearly comprehensive population data to be used meaningfully.