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The most important part of Bayes rule is the concept of Prior knowledge and Bayesian updating - new data should always be combined with your existing knowledge (starting with the "base rate"), based on levels of uncertainty.

Closely related is the concept of conditional probability - how does the probability of something change as you include other information.

It takes some studying to understand probability theory well, but it is very powerful once you start to think that way. Most of the garbage science reporting in the media would be fixed if reporters actually thought this way and honestly applied it.