Rationality

Steven Pinker's Rationality introduces tools for evaluating beliefs and choices, including logic, probability, and decision theory. The author's book listings describe its wider argument about reasoning in society. The worked problem here is Nuxflo's own illustration of why a probability needs a reference population.

Interpret an automated flag

A fictional mail filter flags 90% of unwanted messages and mistakenly flags 10% of ordinary messages. In a batch of 1,000 messages, only 100 are unwanted. What fraction of flagged messages are actually unwanted?

Write counts before reaching for a formula. Of the 100 unwanted messages, 90 are flagged. Of the 900 ordinary messages, 90 are also flagged. There are 180 flags altogether, and 90 are correct. In this batch, a flagged message has a 50% chance of being unwanted, even though the filter catches 90% of unwanted mail.

Change the starting conditions

Now suppose half the messages are unwanted. The same rates produce 450 correct flags and 50 mistaken flags, so 90% of flagged messages are unwanted. The filter's stated detection rates did not change. Its usefulness for interpreting an individual flag changed because the population did.

The exercise distinguishes two questions: “How often does the filter catch unwanted mail?” and “How often is a flagged message unwanted?” Reversing those conditions gives the wrong answer. A diagram or a small table of counts is often enough to expose the difference.

Probability is not the whole decision

Whether to delete a flagged message also depends on the cost of mistakes. Losing an important ordinary message may be worse than reviewing an unwanted one. A review folder could be preferable to automatic deletion even when the filter is reasonably accurate. State those consequences instead of hiding them inside the word “rational.”

This book suits readers willing to work through explicit reasoning tools. It does not supply everyone's goals or settle value disagreements by calculation. For more examples of checking numbers in ordinary claims, see How Not to Be Wrong.