You've probably encountered the phrase "extraordinary claims require extraordinary evidence." Well, it's not just some pithy saying that relieves you of the burden of having to believe every random statement you might encounter in your life. It's actually a handy summary of a very powerful approach to statistics.
The heart here is something called Bayes' theorem, named after the Reverend Thomas Bayes, who was kicking around cool ideas in the first half of the eighteenth century. It relates some statement you're trying to test ("giant stars die as supernovas", "jelly beans cause cancer", "aliens visit Earth because their star exploded and they're going to steal our jelly beans", etc.) to the individual probabilities of each component (the number of stars you observe dying, the number of people eating jelly beans, and so on).
Of course I'm radically simplifying this, but the key takeaway is that Bayesian statistics folds in your existing knowledge of the problem directly into the math, and provides a way to create outcomes that are an updated view of the world based on any new information or experiments. Thus if you don't have a lot of info already, a quick experiment can start to point you in the right direction. But if there is already a huge amount of knowledge about a relationship, it will take a tremendous amount of good evidence to change that perspective.
Unfortunately while simple to state the application of Bayesian statistics gets really complicated really fast, and it's only recently that we've had the computational power to crunch through the intense calculations involved. Which is good. The more we move away from simple-to-apply but simple-to-abuse methods (*cough* p-values *cough*), the better.
Paul M. Sutter is an astrophysicist at The Ohio State University and the chief scientist at COSI science center. Sutter is also host of Ask a Spaceman and Space Radio, and leads AstroTours around the world.