## What probability textbooks don't tell you

I see two major unsatisfactory features of a traditional first course in mathematical probability.
First
**it does not have much connection with how we perceive chance outside the textbook**

as readily demonstrated by our data on
unprompted references to chance.
Second,
**the math gets in the way of the concepts**.

Of course, a cure is harder than a diagnosis, and I don't claim to have a cure.
It's easier to teach the course as just math -- questions stated with X's and Y's
which have definite answers -- without trying to explain
what it all means outside the textbook.
But here are some examples of what I would like to see in a first course.
### The First Commandment of Probability Theory

**Thou shalt not assume different possibilities are equally likely without some darn good reason**.
Here I mean in a real-world context where you wish to place some credence on a numerical estimate of
a probability
(rather than the toy models setting).
This is surely self-evident, but should be emphasized with examples.

- The youtube Frog riddle video
is a fantasy example to illustrate Bayes rule. But the asserted
P(observation | hypothesis) is unequivocally incorrect,
wrongly assuming different possibilities are equally likely.
- The Doomsday argument
which assumes, bizarrely, that ``you" are a uniform random sample from all humans from the past and future.

### GIGO

The phrase *garbage in, garbage out* now sounds very old-fashioned.
But the fact remains that, if you want a meaningful number as a conclusion about the real world, then you usually need first
to input some actual data from the real world.
The custom of filling textbooks with examples and exercises
using totally made-up numbers
as
illustrated in these examples
creates a kind of moral hazard, that it might be reasonable to do this outside of class.
For instance a common application of Bayes rule involves false positives in medical tests.
It is easy to find real data and
learn to explain the conclusions clearly.
### Why do we care about probabilities, anyway?

Here is my discussion of that question.
But textbook writers rarely appear to have given it much thought beyond repeating what they learned as students.