I would argue that the p-value way of reasoning (probability of something happening given a hypothesis) corresponds to an intuitive way many people reason about probability.

For example, suppose the police are investigating the robbery of a home. They find a stranger's (call him person S) fingerprints all over the house. If S claimed he was innocent, a questioner might ask: "If you're really innocent, then why were your fingerprints all over the house?" This is another way of stating that P(Data | innocent) is small (similar to the p-value concept).

Since we often don't have a strong prior reason for believing a given individual is innocent or guilty, the p-value heuristic can be reasonable.

The formal connection to P(innocent | Data) is given by Bayes' theorem