The Dawn Treader

Kudos to Tom for taking a swing at the Bayes problem.

Now that you learned Bayes Theorem here, time to put it to use.

Here's a situation that doctors might encounter:

1% of women at age forty who participate in routine screening have breast cancer.  80% of women with breast cancer will get positive mammographies.  9.6% of women without breast cancer will also get positive mammographies.  A woman in this age group had a positive mammography in a routine screening.  What is the probability that she actually has breast cancer?

What do you think the answer is?

WARNING: Most doctors get the same wrong answer on this problem - usually, only around 15% of doctors get it right.  See Casscells, Schoenberger, and Grayboys 1978; Eddy 1982; Gigerenzer and Hoffrage 1995; and many other studies. 

If you get it right, you are smarter than 85 percent of the doctors in the United States!

Mention the word conditional probability and most people's eyes start to get a glazed look.  So, let's dispense with statistics-speak and splain Bayes Theorem in plain English.  In truth, you already know the gist of Bayes Theorem because it reflects the way we intuitively process observational evidence.

WARNING: this post contains mathematical concepts.

Meet Reverend Thomas Bayes.  He was a Presbyterian minister who lived in London and had a keen interest in mathematics.

He died in 1764, but a formula he developed 300 years ago continues to rock the world ... from search engines to medical diagnosis to spam filters to Christian apologetics.  If you use Google, like most of us do, you are using a Bayesian search engine.  Your spam filters, most likely, are based on a Bayesian approach.