## Presenting probability via math puzzles is harmful

We are all familiar with "mathematical puzzles", exemplified by islands with liars and truth-tellers
You meet two people A and B. A says "Either I am a liar, or B is a truth-teller". What class of person is B?
As intellectual exercises that some people find enjoyable, and that (presented to high school or undergraduate students) may entice some to study mathematics, these are perfectly fine, as are chess problems, Rubik's cube, and so on. But in all of those instances it is clear that one is merely playing an intellectual game, not directly studying the real world. On this page I argue that presenting or teaching probability by using (more than occasionally) such explicit "puzzles", or indeed using the style of unrealistic hypothetical examples that pervade even the best textbooks, is harmful, in the sense of a confusing distraction from understanding the way we really do encounter chance in the real world.

Let me take as an example the puzzle

I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?
discussed in a recent article by science journalist Julie Rehmeyer. As an intellectual puzzle this is (to quote Sherlock Holmes) not entirely devoid of interest. To solve it requires two steps, hopefully feasible to a mathematically-inclined undergraduate.
• decide on a mathematical model you think corresponds to the verbal sentence,
• and do the math calculation within the model.
One ends (within the "most natural" mathematical model) with the counter-intuitive conclusion that the information "born on a Tuesday" makes a difference.

But the major feature of the puzzle is surely that it is no more realistic than islands of liars or Sherlock Holmes or talking rabbits. Real life presents us with many situations where we need or want to make a decision under uncertainty or under partial information -- but one never gets information in the format of the puzzle. People often talk about their children but not in this way! See a twins example for my best attempt to formulate and analyze a more realistic example.

Why do I regard puzzles as so harmful, in the context of probability? Well, my view of the relevant big picture is

The focus in teaching probability should be on thinking seriously about modeling the real world, not on translating word problems into math problems and solving them. Of course this is easier said than done .........