But at some point in understanding probability, as it is actually used in the real world, you need calculus. The same is true for economics, which you didn't mention but often comes up as something everyone should learn. I think by not covering calculus, students are forced to learn things (economics, physics, statistics) the hard way and without understanding them deeply. Unfortunately, calc generally requires trig. It should be taught while teaching calculus.
But at some point in understanding probability, as it is actually used in the real world, you need calculus.
At SOME point, true. But that point is very, very far off, and very few people ever get there.
It is possible to understand what the normal distribution is, and how to lookup significance values, without worrying about how to calculate it. This is how virtually every non-statistician does it, and is what statisticians themselves do more of the time.
The point where you have to calculate it and prove properties about it does require Calculus.
Unfortunately, calc generally requires trig. It should be taught while teaching calculus.
I'm sorry, but this is not a good idea. What Calculus requires from trig is a solid understanding of what sin, cos, and tan are (else the derivatives of the same won't make sense), and a solid understanding of trig identities for use in integration. Both uses require several layers of abstraction on top of the idea of trig. My experience is that it is a bad idea to layer abstractions on top of material before making sure that that material is solid.
I therefore want people to have a solid understanding of trig before they arrive in a Calculus course.
That said, I think that a lot of the use of trig in integration is a now useless skill, given the widespread availability of programs like Mathematica that can solve all of those problems very quickly and much more accurately. It was once important for people to learn those skills, but now it doesn't seem that useful to me.
You should be able to verify that the integral differentiates correctly. But integrating complicated expressions isn't in my view that critical of a skill.
I come from my own point of view that I have difficulty learning something when it is not likely that I will apply it. I learned basic calculus while learning physics, and it was immediately apparent why calculus was required. Without calculus, the problem domain available in physics is almost so trivial that it feels worthless to study.
Basic trig does have other interesting applications, such as computer graphics and what not, that could probably all be integrated into an interesting high school course. So much of a trig class is about memorizing identities, though, and it's hard to see why you might want to do that unless you had something else to do, like integrals. We end up re-learning trig in Calc 2.
I have heard arguments that setting calculus as a "goal" of math education leaves out the stuff of math that is actually "cool" and makes math seem boring. But calculus is interesting (at least) when you use its applications.
Mathematica can, indeed, solve a lot of problems, but is something lost when students never gain the ability to symbolically solve math problems? I suppose there is a wide space for research in this area, and I haven't heard of much being done.
