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.