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Hank: How would you like to be able to tell the future?
Well, you can't. But billions of dollars and millions of lives depend on scientists predicting the future outcome of events every day. From the results of elections to the paths of hurricanes to the likelihood of terrorist attacks, we rely on predictions based on statistical analysis to help us asses risks asses our money and make potentially life altering decisions.

Hank: But how do you determine the odds of something that is inherently uncertain?

Hank: Here's a thought exercise for you: imagine a women  in her 40's who wants to know if she has breast cancer. If she does statistics show there's a 75% chance that a mammogram will detect it. But if she doesn't then there's a 10% chance that the mammogram will give her a false positive. That is, it will say she cancer, and she doesn't. Add this to the fact that we also know that 1.4% of all women in their 40's have breast cancer. Now, imagine our hypothetical women gets a mammogram and it comes back positive. What are the odds that the test is right? You're probably thinking about 75%, right? Wrong. The odds that she actually has cancer are only 9.6%. Now don't feel bad, most doctors get this wrong too.

Hank: This is a problem that calls for Bayesian Reasoning, which, while really counter-intuitive, is the basis for a lot of modern scientific thinking. Bayesian Reasoning is the method by which you can determine the importance of a particular piece of information when trying to prove of disprove a hypothesis. To do that Bayes's Theorem asks for three things:
First, the probability that your hypothesis was true before you conducted the experiment. In this case, our hypothesis is that the woman has cancer when she gets the mammogram.
Second, you have to weigh the odds of a given outcome if your hypothesis is true. In our example, the odds of getting a positive test result.
And third, the odds of that same outcome if your hypothesis is false, meaning the woman gets a positive test but doesn't have cancer.
If you, like most doctors the first time they read this question, thought that a positive test result meant a 75% chance of cancer, you made the very human mistake of thinking that the odds of a woman with cancer getting a positive result were the same as the odds of a woman with a positive result having cancer.

Hank: But they're not! They are in fact totally, drastically different. To think in Bayesian terms is to think this way:
Say you got a group of 1000 women in their 40s. 14 of them have cancer because said that 1.4% of women in this age group have cancer, remember? That means 986 of them do not have cancer. Of the 14 who have cancer, as we established, 75% of them will test positive in a mammogram. 75% of 14 is 10.5 And of the 986 who don't have cancer, 10% of them will receive a positive test result. And 10% of 986 is 98.6. So, after administering 1000 tests, there will be 10.5 true positives and 98.6 false positives. The total number of positive is 109.1. So 10.5 true positive out of 109.1 total positives means that there's a 9.6% chance that a woman in her 40s who receives a positive mammogram result will actually have cancer.

Hank: Thinking like this does not come naturally to most of us, but it's the only way to measure the real significance of experimental results. If you're not thinking like a Bayesian, you can wildly overestimate the relevance of your findings, and that is a big problem in science. Consider for example that 80% of experimental results published in academic journals in the United States turn out to be impossible to replicate. Bayesian Reasoning makes us think rationally, which humans are not always great at. And better reasoning means better science.

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