Bayseian probability seems quite obvious once it is explained, but it is incredibly counterintuitive. Using Bayesian methods is one of the “thinking slow” methods that Daniel Kahneman in Thinking Fast and Slow identifies – very few people can do it intuitively.
So via Barry Ritholtz, here is a great example of why it matters from Nature:
Go ahead and read the whole article. Working out what is likely to be really going on in an experiment is a whole lot trickier than just using a P value based on the size of your sample.